Tutorials in Probability

Tutorials in Probability - Tutorial 1: Dynkin systems 1 1....

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Unformatted text preview: Tutorial 1: Dynkin systems 1 1. Dynkin systems Definition 1 A dynkin system on a set Ω is a subset D of the power set P (Ω), with the following properties: (i) (ii) Ω∈D A, B ∈ D, A ⊆ B ⇒ B \ A ∈ D +∞ (iii) An ∈ D, An ⊆ An+1 , n ≥ 1 ⇒ An ∈ D n=1 Definition 2 A σ -algebra on a set Ω is a subset F of the power set P (Ω) with the following properties: (i) (ii) Ω∈F A ∈ F ⇒ Ac = Ω \ A ∈ F +∞ (iii) An ∈ F , n ≥ 1 ⇒ An ∈ F n=1 Tutorial 1: Dynkin systems 2 Exercise 1. Let F be a σ -algebra on Ω. Show that ∅ ∈ F , that if A, B ∈ F then A ∪ B ∈ F and also A ∩ B ∈ F . Recall that B \ A = B ∩ Ac and conclude that F is also a dynkin system on Ω. Exercise 2. Let (Di )i∈I be an arbitrary family of dynkin systems on Ω, with I = ∅. Show that D = ∩i∈I Di is also a dynkin system on Ω. Exercise 3. Let (Fi )i∈I be an arbitrary family of σ -algebras on Ω, with I = ∅. Show that F = ∩i∈I Fi is also a σ -algebra on Ω. Exercise 4. Let A be a subset of the power set P (Ω). Define: D(A) = {D dynkin system on Ω : A ⊆ D} Show that P (Ω) is a dynkin system on Ω, and that D(A) is not empty. Define: D D(A) = D ∈D(A) Tutorial 1: Dynkin systems 3 Show that D(A) is a dynkin system on Ω such that A ⊆ D(A), and that it is the smallest dynkin system on Ω with such property, (i.e. if D is a dynkin system on Ω with A ⊆ D, then D(A) ⊆ D). Definition 3 Let A ⊆ P (Ω). We call dynkin system generated by A, the dynkin system on Ω, denoted D(A), equal to the intersection of all dynkin systems on Ω, which contain A. Exercise 5. Do exactly as before, but replacing dynkin systems by σ -algebras. Definition 4 Let A ⊆ P (Ω). We call σ -algebra generated by A, the σ -algebra on Ω, denoted σ (A), equal to the intersection of all σ -algebras on Ω, which contain A. Definition 5 A subset A of the power set P (Ω) is called a π -system on Ω, if and only if it is closed under finite intersection, i.e. if it has the property: A, B ∈ A ⇒ A ∩ B ∈ A Tutorial 1: Dynkin systems 4 Exercise 6. Let A be a π -system on Ω. For all A ∈ D(A), we define: Γ(A) = {B ∈ D(A) : A ∩ B ∈ D(A)} 1. If A ∈ A, show that A ⊆ Γ(A) 2. Show that for all A ∈ D(A), Γ(A) is a dynkin system on Ω. 3. Show that if A ∈ A, then D(A) ⊆ Γ(A). 4. Show that if B ∈ D(A), then A ⊆ Γ(B ). 5. Show that for all B ∈ D(A), D(A) ⊆ Γ(B ). 6. Conclude that D(A) is also a π -system on Ω. Exercise 7. Let D be a dynkin system on Ω which is also a π -system. 1. Show that if A, B ∈ D then A ∪ B ∈ D. Tutorial 1: Dynkin systems 2. Let An ∈ D, n ≥ 1. ∞ ∞ ∪+=1 An = ∪+=1 Bn . n n 5 Consider Bn = ∪n Ai . i=1 Show that 3. Show that D is a σ -algebra on Ω. Exercise 8. Let A be a π -system on Ω. Explain why D(A) is a σ -algebra on Ω, and σ (A) is a dynkin system on Ω. Conclude that D(A) = σ (A). Prove the theorem: Theorem 1 (dynkin system) Let C be a collection of subsets of Ω which is closed under pairwise intersection. If D is a dynkin system containing C then D also contains the σ -algebra σ (C ) generated by C . Tutorial 2: Caratheodory’s Extension 1 2. Caratheodory’s Extension In the following, Ω is a set. Whenever a union of sets is denoted as opposed to ∪, it indicates that the sets involved are pairwise disjoint. Definition 6 A semi-ring on Ω is a subset S of the power set P (Ω) with the following properties: (i) (ii) ∅∈S A, B ∈ S ⇒ A ∩ B ∈ S n (iii) A, B ∈ S ⇒ ∃n ≥ 0, ∃Ai ∈ S : A \ B = Ai i=1 The last property (iii) says that whenever A, B ∈ S , there is n ≥ 0 and A1 , . . . , An in S which are pairwise disjoint, such that A \ B = A1 . . . An . If n = 0, it is understood that the corresponding union is equal to ∅, (in which case A ⊆ B ). Tutorial 2: Caratheodory’s Extension 2 Definition 7 A ring on Ω is a subset R of the power set P (Ω) with the following properties: (i) (ii) ∅∈R A, B ∈ R A ∪ B ∈ R (iii) A, B ∈ R ⇒ A \ B ∈ R Exercise 1. Show that A ∩ B = A \ (A \ B ) and therefore that a ring is closed under pairwise intersection. Exercise 2.Show that a ring on Ω is also a semi-ring on Ω. Exercise 3.Suppose that a set Ω can be decomposed as Ω = A1 A2 A3 where A1 , A2 and A3 are distinct from ∅ and Ω. Define S1 = {∅, A1 , A2 , A3 , Ω} and S2 = {∅, A1 , A2 A3 , Ω}. Show that S1 and S2 are semi-rings on Ω, but that S1 ∩ S2 fails to be a semi-ring on Ω. Exercise 4. Let (Ri )i∈I be an arbitrary family of rings on Ω, with I = ∅. Show that R = ∩i∈I Ri is also a ring on Ω. Tutorial 2: Caratheodory’s Extension 3 Exercise 5. Let A be a subset of the power set P (Ω). Define: R(A) = {R ring on Ω : A ⊆ R} Show that P (Ω) is a ring on Ω, and that R(A) is not empty. Define: R R(A) = R∈R(A) Show that R(A) is a ring on Ω such that A ⊆ R(A), and that it is the smallest ring on Ω with such property, (i.e. if R is a ring on Ω and A ⊆ R then R(A) ⊆ R). Definition 8 Let A ⊆ P (Ω). We call ring generated by A, the ring on Ω, denoted R(A), equal to the intersection of all rings on Ω, which contain A. Exercise 6.Let S be a semi-ring on Ω. Define the set R of all finite unions of pairwise disjoint elements of S , i.e. R = {A : A = n i=1 Ai for some n ≥ 0, Ai ∈ S} Tutorial 2: Caratheodory’s Extension 4 (where if n = 0, the corresponding union is empty, i.e. ∅ ∈ R). Let A = n Ai and B = p=1 Bj ∈ R: i=1 j 1. Show that A ∩ B = pairwise intersection. i,j (Ai ∩ Bj ) and that R is closed under 2. Show that if p ≥ 1 then A \ B = ∩p=1 ( j n i=1 (Ai \ Bj )). 3. Show that R is closed under pairwise difference. 4. Show that A ∪ B = (A \ B ) on Ω. 5. Show that R(S ) = R. B and conclude that R is a ring Tutorial 2: Caratheodory’s Extension 5 Exercise 7. Everything being as before, define: R = {A : A = ∪n Ai for some n ≥ 0, Ai ∈ S} i=1 (We do not require the sets involved in the union to be pairwise disjoint). Using the fact that R is closed under finite union, show that R ⊆ R, and conclude that R = R = R(S ). Definition 9 Let A ⊆ P (Ω) with ∅ ∈ A. We call measure on A, any map µ : A → [0, +∞] with the following properties: (i) µ(∅) = 0 +∞ (ii) A ∈ A, An ∈ A and A = +∞ An ⇒ µ(A) = n=1 µ(An ) n=1 The indicates that we assume the An ’s to be pairwise disjoint in the l.h.s. of (ii). It is customary to say in view of condition (ii) that a measure is countably additive. Tutorial 2: Caratheodory’s Extension 6 Exercise 8.If A is a σ -algebra on Ω explain why property (ii) can be replaced by: +∞ (ii) An ∈ A and A = +∞ An ⇒ µ(A) = n=1 µ(An ) n=1 Exercise 9. Let A ⊆ P (Ω) with ∅ ∈ A and µ : A → [0, +∞] be a measure on A. 1. Show that if A1 , . . . , An ∈ A are pairwise disjoint and the union A = n Ai lies in A, then µ(A) = µ(A1 ) + . . . + µ(An ). i=1 2. Show that if A, B ∈ A, A ⊆ B and B \ A ∈ A then µ(A) ≤ µ(B ). Exercise 10. Let S be a semi-ring on Ω, and µ : S → [0, +∞] be a measure on S . Suppose that there exists an extension of µ on R(S ), i.e. a measure µ : R(S ) → [0, +∞] such that µ|S = µ. ¯ ¯ Tutorial 2: Caratheodory’s Extension 7 1. Let A be an element of R(S ) with representation A = n Ai i=1 as a finite union of pairwise disjoint elements of S . Show that n µ(A) = i=1 µ(Ai ) ¯ 2. Show that if µ : R(S ) → [0, +∞] is another measure with ¯ ¯ ¯ µ|S = µ, i.e. another extension of µ on R(S ), then µ = µ. ¯ Exercise 11. Let S be a semi-ring on Ω and µ : S → [0, +∞] be a measure. Let A be an element of R(S ) with two representations: p n Ai = A= i=1 Bj j =1 as a finite union of pairwise disjoint elements of S . 1. For i = 1, . . . , n, show that µ(Ai ) = 2. Show that n i=1 µ(Ai ) = p j =1 µ(Bj ) p j =1 µ(Ai ∩ Bj ) Tutorial 2: Caratheodory’s Extension 8 3. Explain why we can define a map µ : R(S ) → [0, +∞] as: ¯ n µ(A) = ¯ (Ai ) i=1 4. Show that µ(∅) = 0. ¯ Exercise 12. Everything being as before, suppose that (An )n≥1 is a sequence of pairwise disjoint elements of R(S ), each An having the representation: pn Ak , n ≥ 1 n An = k=1 as a finite union of disjoint elements of S . Suppose moreover that ∞ A = +=1 An is an element of R(S ) with representation A = p=1 Bj , n j as a finite union of pairwise disjoint elements of S . ∞ 1. Show that for j = 1, . . . , p, Bj = ∪+=1 ∪pn (Ak ∩ Bj ) and n n k=1 +∞ explain why Bj is of the form Bj = m=1 Cm for some sequence (Cm )m≥1 of pairwise disjoint elements of S . Tutorial 2: Caratheodory’s Extension 2. Show that µ(Bj ) = +∞ n=1 pn k=1 9 µ(Ak ∩ Bj ) n 3. Show that for n ≥ 1 and k = 1, . . . , pn , Ak = n 4. Show that µ(Ak ) = n p j =1 p k j =1 (An ∩ Bj ) µ(Ak ∩ Bj ) n 5. Recall the definition of µ of exercise (11) and show that it is a ¯ measure on R(S ). Exercise 13.Prove the following theorem: Theorem 2 Let S be a semi-ring on Ω. Let µ : S → [0, +∞] be a measure on S . There exists a unique measure µ : R(S ) → [0, +∞] ¯ such that µ|S = µ. ¯ Tutorial 2: Caratheodory’s Extension 10 Definition 10 We define an outer-measure on Ω as being any map µ∗ : P (Ω) → [0, +∞] with the following properties: (i) (ii) (iii) µ∗ (∅) = 0 A ⊆ B ⇒ µ∗ (A) ≤ µ∗ (B ) µ∗ +∞ +∞ An n=1 ≤ µ∗ (An ) n=1 Exercise 14. Show that µ∗ (A ∪ B ) ≤ µ∗ (A) + µ∗ (B ), where µ∗ is an outer-measure on Ω and A, B ⊆ Ω. Definition 11 Let µ∗ be an outer-measure on Ω. We define: Σ(µ∗ ) = {A ⊆ Ω : µ∗ (T ) = µ∗ (T ∩ A) + µ∗ (T ∩ Ac ) , ∀T ⊆ Ω} We call Σ(µ∗ ) the σ -algebra associated with the outer-measure µ∗ . Note that the fact that Σ(µ∗ ) is indeed a σ -algebra on Ω, remains to be proved. This will be your task in the following exercises. Tutorial 2: Caratheodory’s Extension 11 Exercise 15. Let µ∗ be an outer-measure on Ω. Let Σ = Σ(µ∗ ) be the σ -algebra associated with µ∗ . Let A, B ∈ Σ and T ⊆ Ω 1. Show that Ω ∈ Σ and Ac ∈ Σ. 2. Show that µ∗ (T ∩ A) = µ∗ (T ∩ A ∩ B ) + µ∗ (T ∩ A ∩ B c ) 3. Show that T ∩ Ac = T ∩ (A ∩ B )c ∩ Ac 4. Show that T ∩ A ∩ B c = T ∩ (A ∩ B )c ∩ A 5. Show that µ∗ (T ∩ Ac ) + µ∗ (T ∩ A ∩ B c ) = µ∗ (T ∩ (A ∩ B )c ) 6. Adding µ∗ (T ∩(A∩B )) on both sides 5., conclude that A∩B ∈ Σ. 7. Show that A ∪ B and A \ B belong to Σ. Exercise 16. Everything being as before, let An ∈ Σ, n ≥ 1. Define B1 = A1 and Bn+1 = An+1 \ (A1 ∪ . . . ∪ An ). Show that the Bn ’s are ∞ ∞ pairwise disjoint elements of Σ and that ∪+=1 An = +=1 Bn . n n Tutorial 2: Caratheodory’s Extension 12 Exercise 17. Everything being as before, show that if B, C ∈ Σ and B ∩ C = ∅, then µ∗ (T ∩ (B C )) = µ∗ (T ∩ B ) + µ∗ (T ∩ C ) for any T ⊆ Ω. Exercise 18.Everything being as before, let (Bn )n≥1 be a sequence of pairwise disjoint elements of Σ, and let B = 1. Explain why N n=1 Bn 2. Show that µ∗ (T ∩ ( +∞ n=1 Bn . Let N ≥ 1. ∈Σ N n=1 Bn )) = 3. Show that µ∗ (T ∩ B c ) ≤ µ∗ (T ∩ ( 4. Show that µ∗ (T ∩ B c ) + +∞ n=1 N n=1 µ∗ (T ∩ Bn ) N c n=1 Bn ) ) µ∗ (T ∩ Bn ) ≤ µ∗ (T ), and: 5. µ∗ (T ) ≤ µ∗ (T ∩ B c )+ µ∗ (T ∩ B ) ≤ µ∗ (T ∩ B c )+ 6. Show that B ∈ Σ and µ∗ (B ) = +∞ n=1 +∞ n=1 µ∗ (T ∩ Bn ) µ∗ (Bn ). 7. Show that Σ is a σ -algebra on Ω, and µ∗Σ is a measure on Σ. | Tutorial 2: Caratheodory’s Extension 13 Theorem 3 Let µ∗ : P (Ω) → [0, +∞] be an outer-measure on Ω. Then Σ(µ∗ ), the so-called σ -algebra associated with µ∗ , is indeed a σ -algebra on Ω and µ∗Σ(µ∗ ) , is a measure on Σ(µ∗ ). | Exercise 19. Let R be a ring on Ω and µ : R → [0, +∞] be a measure on R. For all T ⊆ Ω, define: µ∗ (T ) = inf +∞ µ(An ) , (An ) is an R-cover of T n=1 where an R-cover of T is defined as any sequence (An )n≥1 of elements ∞ of R such that T ⊆ ∪+=1 An . By convention inf ∅ = + ∞. n 1. Show that µ∗ (∅) = 0. 2. Show that if A ⊆ B then µ∗ (A) ≤ µ∗ (B ). 3. Let (An )n≥1 be a sequence of subsets of Ω, with µ∗ (An ) < +∞ for all n ≥ 1. Given > 0, show that for all n ≥ 1, there exists Tutorial 2: Caratheodory’s Extension 14 an R-cover (Ap )p≥1 of An such that: n +∞ µ(Ap ) < µ∗ (An ) + /2n n p=1 Why is it important to assume µ∗ (An ) < +∞. ∞ 4. Show that there exists an R-cover (Rk ) of ∪+=1 An such that: n +∞ +∞ +∞ µ(Ap ) n µ(Rk ) = k=1 ∞ 5. Show that µ∗ (∪+=1 An ) ≤ + n n=1 p=1 +∞ n=1 µ∗ (An ) 6. Show that µ∗ is an outer-measure on Ω. Tutorial 2: Caratheodory’s Extension 15 Exercise 20. Everything being as before, Let A ∈ R. Let (An )n≥1 be an R-cover of A and put B1 = A1 ∩ A, and: Bn+1 = (An+1 ∩ A) \ ((A1 ∩ A) ∪ . . . ∪ (An ∩ A)) 1. Show that µ∗ (A) ≤ µ(A). 2. Show that (Bn )n≥1 is a sequence of pairwise disjoint elements ∞ of R such that A = +=1 Bn . n 3. Show that µ(A) ≤ µ∗ (A) and conclude that µ∗ = µ. |R Exercise 21. Everything being as before, Let A ∈ R and T ⊆ Ω. 1. Show that µ∗ (T ) ≤ µ∗ (T ∩ A) + µ∗ (T ∩ Ac ). 2. Let (Tn ) be an R-cover of T . Show that (Tn ∩ A) and (Tn ∩ Ac ) are R-covers of T ∩ A and T ∩ Ac respectively. 3. Show that µ∗ (T ∩ A) + µ∗ (T ∩ Ac ) ≤ µ∗ (T ). Tutorial 2: Caratheodory’s Extension 16 4. Show that R ⊆ Σ(µ∗ ). 5. Conclude that σ (R) ⊆ Σ(µ∗ ). Exercise 22.Prove the following theorem: Theorem 4 (caratheodory’s extension) Let R be a ring on Ω and µ : R → [0, +∞] be a measure on R. There exists a measure µ : σ (R) → [0, +∞] such that µ|R = µ. Exercise 23. Let S be a semi-ring on Ω. Show that σ (R(S )) = σ (S ). Exercise 24.Prove the following theorem: Theorem 5 Let S be a semi-ring on Ω and µ : S → [0, +∞] be a measure on S . There exists a measure µ : σ (S ) → [0, +∞] such that µ|S = µ. Tutorial 3: Stieltjes-Lebesgue Measure 1 3. Stieltjes-Lebesgue Measure Definition 12 Let A ⊆ P (Ω) and µ : A → [0, +∞] be a map. We say that µ is finitely additive if and only if, given n ≥ 1: n A ∈ A, Ai ∈ A, A = n Ai ⇒ µ(A) = i=1 µ(Ai ) i=1 We say that µ is finitely sub-additive if and only if, given n ≥ 1 : n A ∈ A, Ai ∈ A, A ⊆ n Ai ⇒ µ(A) ≤ i=1 µ(Ai ) i=1 Exercise 1. Let S = {]a, b] , a, b ∈ R} be the set of all intervalsa, b], defined as ]a, b] = {x ∈ R, a < x ≤ b}. 1. Show that ]a, b]∩]c, d] =]a ∨ c, b ∧ d] 2. Show that ]a, b]\]c, d] =]a, b ∧ c]∪]a ∨ d, b] Tutorial 3: Stieltjes-Lebesgue Measure 2 3. Show that c ≤ d ⇒ b ∧ c ≤ a ∨ d. 4. Show that S is a semi-ring on R. Exercise 2. Suppose S is a semi-ring in Ω and µ : S → [0, +∞] is finitely additive. Show that µ can be extended to a finitely additive map µ : R(S ) → [0, +∞], with µ|S = µ. ¯ ¯ Exercise 3. Everything being as before, Let A ∈ R(S ), Ai ∈ R(S ), A ⊆ ∪n Ai where n ≥ 1. Define B1 = A1 ∩ A and for i = 1, . . . , n − 1: i=1 Bi+1 = (Ai+1 ∩ A) \ ((A1 ∩ A) ∪ . . . ∪ (Ai ∩ A)) 1. Show that B1 , . . . , Bn are pairwise disjoint elements of R(S ) such that A = n Bi . i=1 ¯ 2. Show that for all i = 1, . . . , n, we have µ(Bi ) ≤ µ(Ai ). ¯ 3. Show that µ is finitely sub-additive. ¯ 4. Show that µ is finitely sub-additive. Tutorial 3: Stieltjes-Lebesgue Measure 3 Exercise 4. Let F : R → R be a right-continuous, non-decreasing map. Let S be the semi-ring on R, S = {]a, b] , a, b ∈ R}. Define the map µ : S → [0, +∞] by µ(∅) = 0, and: ∀a ≤ b , µ(]a, b]) = F (b) − F (a) (1) Let a < b and ai < bi for i = 1, . . . , n and n ≥ 1, with : n a, b] = ai , bi ] i=1 1. Show that there is i1 ∈ {1, . . . , n} such that ai1 = a. 2. Show that ]bi1 , b] = i∈{1,...,n}\{i1 } ]ai , bi ] 3. Show the existence of a permutation (i1 , . . . , in ) of {1, . . . , n} such that a = ai1 < bi1 = ai2 < . . . < bin = b. 4. Show that µ is finitely additive and finitely sub-additive. Tutorial 3: Stieltjes-Lebesgue Measure 4 Exercise 5. µ being defined as before, suppose a < b and an < bn for n ≥ 1 with: +∞ an , bn ] a, b] = n=1 Given N ≥ 1, let (i1 , . . . , iN ) be a permutation of {1, . . . , N } with: a ≤ ai1 < bi1 ≤ ai2 < . . . < biN ≤ b 1. Show that N k=1 F (bik ) − F (aik ) ≤ F (b) − F (a). 2. Show that +∞ n=1 µ(]an , bn ]) ≤ µ(]a, b]) 3. Given > 0, show that there is η ∈]0, b − a[ such that: 0 ≤ F (a + η ) − F (a) ≤ 4. For n ≥ 1, show that there is ηn > 0 such that: 0 ≤ F (bn + ηn ) − F (bn ) ≤ 2n Tutorial 3: Stieltjes-Lebesgue Measure 5 ∞ 5. Show that [a + η, b] ⊆ ∪+=1 ]an , bn + ηn [. n 6. Explain why there exist p ≥ 1 and integers n1 , . . . , np such that:a + η, b] ⊆ ∪p=1 ]ank , bnk + ηnk ] k 7. Show that F (b) − F (a) ≤ 2 + +∞ n=1 F (bn ) − F (an ) 8. Show that µ : S → [0, +∞] is a measure. Definition 13 A topology on Ω is a subset T of the power set P (Ω), with the following properties: (i) (ii) (iii) Ω, ∅ ∈ T A, B ∈ T ⇒ A ∩ B ∈ T Ai ∈ T , ∀i ∈ I ⇒ Ai ∈ T i∈I Tutorial 3: Stieltjes-Lebesgue Measure 6 Property (iii) of definition (13) can be translated as: for any family (Ai )i∈I of elements of T , the union ∪i∈I Ai is still an element of T . Hence, a topology on Ω, is a set of subsets of Ω containing Ω and the empty set, which is closed under finite intersection and arbitrary union. Definition 14 A topological space is an ordered pair (Ω, T ), where Ω is a set and T is a topology on Ω. Definition 15 Let (Ω, T ) be a topological space. We say that A ⊆ Ω is an open set in Ω, if and only if it is an element of the topology T . We say that A ⊆ Ω is a closed set in Ω, if and only if its complement Ac is an open set in Ω. Definition 16 Let (Ω, T ) be a topological space. We define the borel σ -algebra on Ω, denoted B (Ω), as the σ -algebra on Ω, generated by the topology T . In other words, B (Ω) = σ (T ) Tutorial 3: Stieltjes-Lebesgue Measure 7 Definition 17 We define the usual topology on R, denoted TR , as the set of all U ⊆ R such that: ∀x ∈ U , ∃ > 0 , ]x − , x + [⊆ U Exercise 6.Show that TR is indeed a topology on R. Exercise 7. Consider the semi-ring S = {]a, b] , a, b ∈ R}. Let TR be the usual topology on R, and B (R) be the borel σ -algebra on R. ∞ 1. Let a ≤ b. Show that ]a, b] = ∩+=1 ]a, b + 1/n[. n 2. Show that σ (S ) ⊆ B (R). 3. Let U be an open subset of R. Show that for all x ∈ U , there exist ax , bx ∈ Q such that x ∈]ax , bx ] ⊆ U . 4. Show that U = ∪x∈U ]ax , bx ]. 5. Show that the set I = {]ax , bx ] , x ∈ U } is countable. Tutorial 3: Stieltjes-Lebesgue Measure 8 6. Show that U can be written U = ∪i∈I Ai with Ai ∈ S . 7. Show that σ (S ) = B (R). Theorem 6 Let S be the semi-ring S = {]a, b] , a, b ∈ R}. Then, the borel σ -algebra B (R) on R, is generated by S , i.e. B (R) = σ (S ). Definition 18 A measurable space is an ordered pair (Ω, F ) where Ω is a set and F is a σ -algebra on Ω. Definition 19 A measure space is a triple (Ω, F , µ) where (Ω, F ) is a measurable space and µ : F → [0, +∞] is a measure on F . Tutorial 3: Stieltjes-Lebesgue Measure 9 Exercise 8.Let (Ω, F , µ) be a measure space. Let (An )n≥1 be a sequence of elements of F such that An ⊆ An+1 for all n ≥ 1, and let ∞ A = ∪+=1 An (we write An ↑ A). Define B1 = A1 and for all n ≥ 1, n Bn+1 = An+1 \ An . 1. Show that (Bn ) is a sequence of pairwise disjoint elements of F ∞ such that A = +=1 Bn . n 2. Given N ≥ 1 show that AN = N n=1 Bn . 3. Show that µ(AN ) → µ(A) as N → +∞ 4. Show that µ(An ) ≤ µ(An+1 ) for all n ≥ 1. Theorem 7 Let (Ω, F , µ) be a measure space. Then if (An )n≥1 is a sequence of elements of F , such that An ↑ A, we have µ(An ) ↑ µ(A)1 . 1 i.e. the sequence (µ(An ))n≥1 is non-decreasing and converges to µ(A). Tutorial 3: Stieltjes-Lebesgue Measure 10 Exercise 9.Let (Ω, F , µ) be a measure space. Let (An )n≥1 be a sequence of elements of F such that An+1 ⊆ An for all n ≥ 1, and let ∞ A = ∩+=1 An (we write An ↓ A). We assume that µ(A1 ) < +∞. n 1. Define Bn = A1 \ An and show that Bn ∈ F , Bn ↑ A1 \ A. 2. Show that µ(Bn ) ↑ µ(A1 \ A) 3. Show that µ(An ) = µ(A1 ) − µ(A1 \ An ) 4. Show that µ(A) = µ(A1 ) − µ(A1 \ A) 5. Why is µ(A1 ) < +∞ important in deriving those equalities. 6. Show that µ(An ) → µ(A) as n → +∞ 7. Show that µ(An+1 ) ≤ µ(An ) for all n ≥ 1. Theorem 8 Let (Ω, F , µ) be a measure space. Then if (An )n≥1 is a sequence of elements of F , such that An ↓ A and µ(A1 ) < +∞, we have µ(An ) ↓ µ(A). Tutorial 3: Stieltjes-Lebesgue Measure 11 Exercise 10.Take Ω = R and F = B (R). Suppose µ is a measure on B (R) such that µ(]a, b]) = b − a, for a < b. Take An =]n, +∞[. 1. Show that An ↓ ∅. 2. Show that µ(An ) = +∞, for all n ≥ 1. 3. Conclude that µ(An ) ↓ µ(∅) fails to be true. Exercise 11. Let F : R → R be a right-continuous, non-decreasing map. Show the existence of a measure µ : B (R) → [0, +∞] such that: ∀a, b ∈ R , a ≤ b , µ(]a, b]) = F (b) − F (a) (2) Exercise 12.Let µ1 , µ2 be two measures on B (R) with property (2). For n ≥ 1, we define: Dn = {B ∈ B (R) , µ1 (B ∩] − n, n]) = µ2 (B ∩] − n, n])} 1. Show that Dn is a dynkin system on R. Tutorial 3: Stieltjes-Lebesgue Measure 12 2. Explain why µ1 (] − n, n]) < +∞ and µ2 (] − n, n]) < +∞ is needed when proving 1. 3. Show that S = {]a, b] , a, b ∈ R} ⊆ Dn . 4. Show that B (R) ⊆ Dn . 5. Show that µ1 = µ2 . 6. Prove the following theorem. Theorem 9 Let F : R → R be a right-continuous, non-decreasing map. There exists a unique measure µ : B (R) → [0, +∞] such that: ∀a, b ∈ R , a ≤ b , µ(]a, b]) = F (b) − F (a) Definition 20 Let F : R → R be a right-continuous, non-decreasing map. We call stieltjes measure on R associated with F , the unique measure on B (R), denoted dF , such that: ∀a, b ∈ R , a ≤ b , dF (]a, b]) = F (b) − F (a) Tutorial 3: Stieltjes-Lebesgue Measure 13 Definition 21 We call lebesgue measure on R, the unique measure on B (R), denoted dx, such that: ∀a, b ∈ R , a ≤ b , dx(]a, b]) = b − a Exercise 13. Let F : R → R be a right-continuous, non-decreasing map. Let x0 ∈ R. 1. Show that the limit F (x0 −) = limx<x0 ,x→x0 F (x) exists and is an element of R. ∞ 2. Show that {x0 } = ∩+=1 ]x0 − 1/n, x0 ]. n 3. Show that {x0 } ∈ B (R) 4. Show that dF ({x0 }) = F (x0 ) − F (x0 −) Tutorial 3: Stieltjes-Lebesgue Measure 14 Exercise 14.Let F : R → R be a right-continuous, non-decreasing map. Let a ≤ b. 1. Show that ]a, b] ∈ B (R) and dF (]a, b]) = F (b) − F (a) 2. Show that [a, b] ∈ B (R) and dF ([a, b]) = F (b) − F (a−) 3. Show that ]a, b[∈ B (R) and dF (]a, b[) = F (b−) − F (a) 4. Show that [a, b[∈ B (R) and dF ([a, b[) = F (b−) − F (a−) Exercise 15. Let A be a subset of the power set P (Ω). Let Ω ⊆ Ω. Define: A|Ω = {A ∩ Ω , A ∈ A} 1. Show that if A is a topology on Ω, A|Ω is a topology on Ω’. 2. Show that if A is a σ -algebra on Ω, A|Ω is a σ -algebra on Ω’. Tutorial 3: Stieltjes-Lebesgue Measure 15 Definition 22 Let Ω be a set, and Ω ⊆ Ω. Let A be a subset of the power set P (Ω). We call trace of A on Ω’, the subset A|Ω of the power set P (Ω ) defined by: A|Ω = {A ∩ Ω , A ∈ A} Definition 23 Let (Ω, T ) be a topological space and Ω ⊆ Ω. We call induced topology on Ω’, denoted T|Ω , the topology on Ω’ defined by: T|Ω = {A ∩ Ω , A ∈ T } In other words, the induced topology T|Ω is the trace of T on Ω’. Exercise 16.Let A be a subset of the power set P (Ω). Let Ω ⊆ Ω, and A|Ω be the trace of A on Ω’. Define: Γ = {A ∈ σ (A) , A ∩ Ω ∈ σ (A|Ω )} where σ (A|Ω ) refers to the σ -algebra generated by A|Ω on Ω’. Tutorial 3: Stieltjes-Lebesgue Measure 16 1. Explain why the notation σ (A|Ω ) by itself is ambiguous. 2. Show that A ⊆ Γ. 3. Show that Γ is a σ -algebra on Ω. 4. Show that σ (A|Ω ) = σ (A)|Ω Theorem 10 Let Ω ⊆ Ω and A be a subset of the power set P (Ω). Then, the trace on Ω’ of the σ -algebra σ (A) generated by A, is equal to the σ -algebra on Ω’ generated by the trace of A on Ω’. In other words, σ (A)|Ω = σ (A|Ω ). Exercise 17.Let (Ω, T ) be a topological space and Ω ⊆ Ω with its induced topology T|Ω . 1. Show that B (Ω)|Ω = B (Ω ). 2. Show that if Ω ∈ B (Ω) then B (Ω ) ⊆ B (Ω). Tutorial 3: Stieltjes-Lebesgue Measure 17 3. Show that B (R+ ) = {A ∩ R+ , A ∈ B (R)}. 4. Show that B (R+ ) ⊆ B (R). Exercise 18.Let (Ω, F , µ) be a measure space and Ω ⊆ Ω 1. Show that (Ω , F|Ω ) is a measurable space. 2. If Ω ∈ F , show that F|Ω ⊆ F . 3. If Ω ∈ F , show that (Ω , F|Ω , µ|Ω ) is a measure space, where µ|Ω is defined as µ|Ω = µ|(F|Ω ) . Exercise 19. Let F : R+ → R be a right-continuous, non-decreasing map with F (0) ≥ 0. Define: ¯ F (x) = 0 F (x) if if x<0 x≥0 ¯ 1. Show that F : R → R is right-continuous and non-decreasing. Tutorial 3: Stieltjes-Lebesgue Measure 18 ¯ 2. Show that µ : B (R+ ) → [0, +∞] defined by µ = dF|B(R+ ) , is a + measure on B (R ) with the properties: (i) (ii) µ({0}) = F (0) ∀0 ≤ a ≤ b , µ(]a, b]) = F (b) − F (a) Exercise 20. Define: C = {{0}} ∪ {]a, b] , 0 ≤ a ≤ b} 1. Show that C ⊆ B (R+ ) 2. Let U be open in R+ . Show that U is of the form: (R+ ∩]ai , bi ]) U= i∈I where I is a countable set and ai , bi ∈ R with ai ≤ bi . 3. For all i ∈ I , show that R+ ∩]ai , bi ] ∈ σ (C ). 4. Show that σ (C ) = B (R+ ) Tutorial 3: Stieltjes-Lebesgue Measure 19 Exercise 21.Let µ1 and µ2 be two measures on B (R+ ) with: (i) (ii) µ1 ({0}) = µ2 ({0}) = F (0) µ1 (]a, b]) = µ2 (]a, b]) = F (b) − F (a) for all 0 ≤ a ≤ b. For n ≥ 1, we define: Dn = {B ∈ B (R+ ) , µ1 (B ∩ [0, n]) = µ2 (B ∩ [0, n])} 1. Show that Dn is a dynkin system on R+ with C ⊆ Dn , where the set C is defined as in exercise (20). 2. Explain why µ1 ([0, n]) < +∞ and µ2 ([0, n]) < +∞ is important when proving 1. 3. Show that µ1 = µ2 . 4. Prove the following theorem. Tutorial 3: Stieltjes-Lebesgue Measure 20 Theorem 11 Let F : R+→ R be a right-continuous, non-decreasing map with F (0) ≥ 0. There exists a unique µ : B (R+ ) → [0, +∞] measure on B (R+ ) such that: (i) (ii) µ({0}) = F (0) ∀0 ≤ a ≤ b , µ(]a, b]) = F (b) − F (a) Definition 24 Let F : R+→ R be a right-continuous, non-decreasing map with F (0) ≥ 0. We call stieltjes measure on R+ associated with F , the unique measure on B (R+ ), denoted dF , such that: (i) (ii) dF ({0}) = F (0) ∀0 ≤ a ≤ b , dF (]a, b]) = F (b) − F (a) Tutorial 4: Measurability 1 4. Measurability Definition 25 Let A and B be two sets, and f : A → B be a map. Given A ⊆ A, we call direct image of A by f the set denoted f (A ), and defined by f (A ) = {f (x) : x ∈ A }. Definition 26 Let A and B be two sets, and f : A → B be a map. Given B ⊆ B , we call inverse image of B by f the set denoted f −1 (B ), and defined by f −1 (B ) = {x : x ∈ A , f (x) ∈ B }. Exercise 1. Let A and B be two sets, and f : A → B be a bijection from A to B . Let A ⊆ A and B ⊆ B . 1. Explain why the notation f −1 (B ) is potentially ambiguous. 2. Show that the inverse image of B by f is in fact equal to the direct image of B by f −1 . 3. Show that the direct image of A by f is in fact equal to the inverse image of A by f −1 . Tutorial 4: Measurability 2 Definition 27 Let (Ω, T ) and (S, TS ) be two topological spaces. A map f : Ω → S is said to be continuous if and only if: ∀B ∈ TS , f −1 (B ) ∈ T In other words, if and only if the inverse image of any open set in S is an open set in Ω. We Write f : (Ω, T ) → (S, TS ) is continuous, as a way of emphasizing the two topologies T and TS with respect to which f is continuous. Definition 28 Let E be a set. A map d : E × E → [0, +∞[ is said to be a metric on E , if and only if: (i) (ii) (iii) ∀x, y ∈ E , d(x, y ) = 0 ⇔ x = y ∀x, y ∈ E , d(x, y ) = d(y, x) ∀x, y, z ∈ E , d(x, y ) ≤ d(x, z ) + d(z, y ) Definition 29 A metric space is an ordered pair (E, d) where E is a set, and d is a metric on E . Tutorial 4: Measurability 3 Definition 30 Let (E, d) be a metric space. For all x ∈ E and > 0, we define the so-called open ball in E : B (x, ) = {y : y ∈ E , d(x, y ) < } d We call metric topology on E , associated with d, the topology TE defined by: d TE = {U ⊆ E , ∀x ∈ U, ∃ > 0, B (x, ) ⊆ U } d Exercise 2. Let TE be the metric topology associated with d, where (E, d) is a metric space. d 1. Show that TE is indeed a topology on E . 2. Given x ∈ E and > 0, show that B (x, ) is an open set in E . Exercise 3. Show that the usual topology on R is nothing but the metric topology associated with d(x, y ) = |x − y |. Tutorial 4: Measurability 4 Exercise 4. Let (E, d) and (F, δ ) be two metric spaces. Show that a map f : E → F is continuous, if and only if for all x ∈ E and > 0, there exists η > 0 such that for all y ∈ E : d(x, y ) < η ⇒ δ (f (x), f (y )) < Definition 31 Let (Ω, T ) and (S, TS ) be two topological spaces. A map f : Ω → S is said to be a homeomorphism, if and only if f is a continuous bijection, such that f −1 is also continuous. Definition 32 A topological space (Ω, T ) is said to be metrizable, if and only if there exists a metric d on Ω, such that the associated d metric topology coincides with T , i.e. TΩ = T . Tutorial 4: Measurability 5 Definition 33 Let (E, d) be a metric space and F ⊆ E . We call induced metric on F , denoted d|F , the restriction of the metric d to F × F , i.e. d|F = d|F ×F . Exercise 5.Let (E, d) be a metric space and F ⊆ E . We define d TF = (TE )|F as the topology on F induced by the metric topology on d E . Let TF = TF |F be the metric topology on F associated with the induced metric d|F on F . 1. Show that TF ⊆ TF . 2. Given A ∈ TF , show that A = (∪x∈A B (x, x )) ∩ F for some x > 0, x ∈ A, where B (x, x ) denotes the open ball in E . 3. Show that TF ⊆ TF . Theorem 12 Let (E, d) be a metric space and F ⊆ E . Then, the topology on F induced by the metric topology, is equal to the metric d d topology on F associated with the induced metric, i.e. (TE )|F = TF |F . Tutorial 4: Measurability 6 Exercise 6. Let φ : R →] − 1, 1[ be the map defined by: x ∀x ∈ R , φ(x) = |x| + 1 1. Show that [−1, 0[ is not open in R. 2. Show that [−1, 0[ is open in [−1, 1]. 3. Show that φ is a homeomorphism between R and ] − 1, 1[. 4. Show that limx→+∞ φ(x) = 1 and limx→−∞ φ(x) = −1. ¯ Exercise 7. Let R = [−∞, +∞] = R ∪{−∞, +∞}. Let φ be defined ¯¯ as in exercise (6), and φ : R → [−1, 1] be the map defined by: φ(x) if x ∈ R ¯ φ(x) = 1 if x = +∞ −1 if x = −∞ Define: ¯¯ TR = {U ⊆ R , φ(U ) is open in [−1, 1]} ¯ Tutorial 4: Measurability 7 ¯ ¯ ¯¯ 1. Show that φ is a bijection from R to [−1, 1], and let ψ = φ−1 . ¯ 2. Show that TR is a topology on R. ¯ ¯ ¯ 3. Show that φ is a homeomorphism between R and [−1, 1]. ¯ 4. Show that [−∞, 2[, ]3, +∞], ]3, +∞[ are open in R. ¯ 5. Show that if φ : R → [−1, 1] is an arbitrary homeomorphism, ¯ is open, if and only if φ (U ) is open in [−1, 1]. then U ⊆ R ¯ Definition 34 The usual topology on R is defined as: ¯¯ TR = {U ⊆ R , φ(U ) is open in [−1, 1]} ¯ ¯¯ ¯ ¯ where φ : R → [−1, 1] is defined by φ(−∞) = −1, φ(+∞) = 1 and: x ¯ ∀x ∈ R , φ(x) = |x| + 1 Tutorial 4: Measurability 8 ¯ Exercise 8. Let φ and φ be as in exercise (7). Define: T = (TR )|R = {U ∩ R , U ∈ TR } ¯ ¯ 1. Recall why T is a topology on R. ¯ ¯ 2. Show that for all U ⊆ R, φ(U ∩ R) = φ(U )∩] − 1, 1[. 3. Explain why if U ∈ TR , φ(U ∩ R) is open in ] − 1, 1[. ¯ 4. Show that T ⊆ TR , (the usual topology on R). ¯ 5. Let U ∈ TR . Show that φ(U ) is open in ] − 1, 1[ and [−1, 1]. 6. Show that TR ⊆ TR ¯ ¯ 7. Show that TR = T , i.e. that the usual topology on R induces the usual topology on R. ¯ ¯ 8. Show that B (R) = B (R)|R = {B ∩ R , B ∈ B (R)} Tutorial 4: Measurability 9 ¯ ¯ Exercise 9.Let d : R × R → [0, +∞[ be defined by: ¯ ¯ ∀(x, y ) ∈ R × R , d(x, y ) = |φ(x) − φ(y )| ¯ where φ is an arbitrary homeomorphism from R to [−1, 1]. ¯ 1. Show that d is a metric on R. 2. Show that if U ∈ TR , then φ(U ) is open in [−1, 1] ¯ 3. Show that for all U ∈ TR and y ∈ φ(U ), there exists ¯ that: ∀z ∈ [−1, 1] , |z − y | < ⇒ z ∈ φ(U ) > 0 such d 4. Show that TR ⊆ TR . ¯ ¯ d 5. Show that for all U ∈ TR and x ∈ U , there is ¯ ¯ ∀y ∈ R , |φ(x) − φ(y )| < > 0 such that: ⇒ y∈U d 6. Show that for all U ∈ TR , φ(U ) is open in [−1, 1]. ¯ Tutorial 4: Measurability 10 d 7. Show that TR ⊆ TR ¯ ¯ 8. Prove the following theorem. ¯¯ Theorem 13 The topological space (R, TR ) is metrizable. Definition 35 Let (Ω, F ) and (S, Σ) be two measurable spaces. A map f : Ω → S is said to be measurable with respect to F and Σ, if and only if: ∀B ∈ Σ , f −1 (B ) ∈ F We Write f : (Ω, F ) → (S, Σ) is measurable, as a way of emphasizing the two σ -algebra F and Σ with respect to which f is measurable. Exercise 10. Let (Ω, F ) and (S, Σ) be two measurable spaces. Let S be a set and f : Ω → S be a map such that f (Ω) ⊆ S ⊆ S . We define Σ as the trace of Σ on S , i.e. Σ = Σ|S . 1. Show that for all B ∈ Σ, we have f −1 (B ) = f −1 (B ∩ S ) Tutorial 4: Measurability 11 2. Show that f : (Ω, F ) → (S, Σ) is measurable, if and only if f : (Ω, F ) → (S , Σ ) is itself measurable. 3. Let f : Ω → R+ . Show that the following are equivalent: (i) (ii) (iii) f : (Ω, F ) → (R+ , B (R+ )) is measurable f : (Ω, F ) → (R, B (R)) is measurable ¯ ¯ f : (Ω, F ) → (R, B (R)) is measurable Exercise 11. Let (Ω, F ), (S, Σ), (S1 , Σ1 ) be three measurable spaces. let f : (Ω, F ) → (S, Σ) and g : (S, Σ) → (S1 , Σ1 ) be two measurable maps. 1. For all B ⊆ S1 , show that (g ◦ f )−1 (B ) = f −1 (g −1 (B )) 2. Show that g ◦ f : (Ω, F ) → (S1 , Σ1 ) is measurable. Tutorial 4: Measurability 12 Exercise 12.Let (Ω, F ) and (S, Σ) be two measurable spaces. Let f : Ω → S be a map. We define: Γ = {B ∈ Σ , f −1 (B ) ∈ F } 1. Show that f −1 (S ) = Ω. 2. Show that for all B ⊆ S , f −1 (B c ) = (f −1 (B ))c . ∞ ∞ 3. Show that if Bn ⊆ S, n ≥ 1, then f −1 (∪+=1 Bn ) = ∪+=1 f −1 (Bn ) n n 4. Show that Γ is a σ -algebra on S . 5. Prove the following theorem. Theorem 14 Let (Ω, F ) and (S, Σ) be two measurable spaces, and A be a set of subsets of S generating Σ, i.e. such that Σ = σ (A). Then f : (Ω, F ) → (S, Σ) is measurable, if and only if: ∀B ∈ A , f −1 (B ) ∈ F Tutorial 4: Measurability 13 Exercise 13. Let (Ω, T ) and (S, TS ) be two topological spaces. Let f : Ω → S be a map. Show that if f : (Ω, T ) → (S, TS ) is continuous, then f : (Ω, B (Ω)) → (S, B (S )) is measurable. ¯ Exercise 14.We define the following subsets of the power set P (R): C1 = {[−∞, c] , c ∈ R} C2 = {[−∞, c[ , c ∈ R} C3 = {[c, +∞] , c ∈ R} C4 = {]c, +∞] , c ∈ R} 1. Show that C2 and C4 are subsets of TR . ¯ ¯ 2. Show that the elements of C1 and C3 are closed in R. ¯ 3. Show that for all i = 1, 2, 3, 4, σ (Ci ) ⊆ B (R). ¯ 4. Let U be open in R. Explain why U ∩ R is open in R. Tutorial 4: Measurability 14 5. Show that any open subset of R is a countable union of open bounded intervals in R. 6. Let a < b, a, b ∈ R. Show that we have: +∞ a, b[= +∞ a, b − 1/n] = n=1 [a + 1/n, b[ n=1 7. Show that for all i = 1, 2, 3, 4, ]a, b[∈ σ (Ci ). 8. Show that for all i = 1, 2, 3, 4, {{−∞}, {+∞}} ⊆ σ (Ci ). ¯ 9. Show that for all i = 1, 2, 3, 4, σ (Ci ) = B (R) 10. Prove the following theorem. Tutorial 4: Measurability 15 ¯ Theorem 15 Let (Ω, F ) be a measurable space, and f : Ω → R be a map. The following are equivalent: (i) (ii) ¯ ¯ f : (Ω, F ) → (R, B (R)) is measurable ¯ ∀B ∈ B (R) , {f ∈ B } ∈ F (iii) (iv ) ∀c ∈ R , {f ≤ c} ∈ F ∀c ∈ R , {f < c} ∈ F (v ) (vi) ∀c ∈ R , {c ≤ f } ∈ F ∀c ∈ R , {c < f } ∈ F Exercise 15. Let (Ω, F ) be a measurable space. Let (fn )n≥1 be a ¯ ¯ sequence of measurable maps fn : (Ω, F ) → (R, B (R)). Let g and h be the maps defined by g (ω ) = inf n≥1 fn (ω ) and h(ω ) = supn≥1 fn (ω ), for all ω ∈ Ω. ∞ 1. Let c ∈ R. Show that {c ≤ g } = ∩+=1 {c ≤ fn }. n ∞ 2. Let c ∈ R. Show that {h ≤ c} = ∩+=1 {fn ≤ c}. n Tutorial 4: Measurability 16 ¯ ¯ 3. Show that g, h : (Ω, F ) → (R, B (R)) are measurable. ¯ Definition 36 Let (vn )n≥1 be a sequence in R. We define: u = lim inf vn = sup n→+∞ n≥1 inf vk k ≥n and: w = lim sup vn = inf n→+∞ n≥1 sup vk k ≥n ¯ Then, u, w ∈ R are respectively called lower limit and upper limit of the sequence (vn )n≥1 . ¯ Exercise 16. Let (vn )n≥1 be a sequence in R. for n ≥ 1 we define un = inf k≥n vk and wn = supk≥n vk . Let u and w be the lower limit and upper limit of (vn )n≥1 , respectively. 1. Show that un ≤ un+1 ≤ u, for all n ≥ 1. Tutorial 4: Measurability 17 2. Show that w ≤ wn+1 ≤ wn , for all n ≥ 1. 3. Show that un → u and wn → w as n → +∞. 4. Show that un ≤ vn ≤ wn , for all n ≥ 1. 5. Show that u ≤ w. ¯ 6. Show that if u = w then (vn )n≥1 converges to a limit v ∈ R, with u = v = w. 7. Show that if a, b ∈ R are such that u < a < b < w then for all n ≥ 1, there exist N1 , N2 ≥ n such that vN1 < a < b < vN2 . 8. Show that if a, b ∈ R are such that u < a < b < w then there exist two strictly increasing sequences of integers (nk )k≥1 and (mk )k≥1 such that for all k ≥ 1, we have vnk < a < b < vmk . ¯ 9. Show that if (vn )n≥1 converges to some v ∈ R, then u = w. Tutorial 4: Measurability 18 ¯ Theorem 16 Let (vn )n≥1 be a sequence in R. Then, the following are equivalent: (i) (ii) lim inf vn = lim sup vn n→+∞ n→+∞ ¯ lim vn exists in R. n→+∞ in which case: lim vn = lim inf vn = lim sup vn n→+∞ n→+∞ n→+∞ ¯ ¯ Exercise 17. Let f, g : (Ω, F ) → (R, B (R)) be two measurable maps, where (Ω, F ) is a measurable space. 1. Show that {f < g } = ∪r∈Q ({f < r} ∩ {r < g }). 2. Show that the sets {f < g }, {f > g }, {f = g }, {f ≤ g }, {f ≥ g } belong to the σ -algebra F . Tutorial 4: Measurability 19 Exercise 18. Let (Ω, F ) be a measurable space. Let (fn )n≥1 be ¯ ¯ a sequence of measurable maps fn : (Ω, F ) → (R, B (R)). We define g = lim inf fn and h = lim sup fn in the obvious way: ∀ω ∈ Ω , g (ω) = lim inf fn (ω ) n→+∞ ∀ω ∈ Ω , h(ω ) = lim sup fn (ω ) n→+∞ ¯ ¯ 1. Show that g, h : (Ω, F ) → (R, B (R)) are measurable. 2. Show that g ≤ h, i.e. ∀ω ∈ Ω , g (ω) ≤ h(ω ). 3. Show that {g = h} ∈ F . ¯ 4. Show that {ω : ω ∈ Ω , limn→+∞ fn (ω ) exists in R} ∈ F . 5. Suppose Ω = {g = h}, and let f (ω) = limn→+∞ fn (ω ), for all ¯ ¯ ω ∈ Ω. Show that f : (Ω, F ) → (R, B (R)) is measurable. Tutorial 4: Measurability 20 ¯ ¯ Exercise 19. Let f, g : (Ω, F ) → (R, B (R)) be two measurable maps, where (Ω, F ) is a measurable space. 1. Show that −f, |f |, f + = max(f, 0) and f − = max(−f, 0) are ¯ measurable with respect to F and B (R). ¯ 2. Let a ∈ R. Explain why the map a + f may not be well defined. ¯ ¯ 3. Show that (a + f ) : (Ω, F ) → (R, B (R)) is measurable, whenever a ∈ R. ¯ ¯ 4. Show that (a.f ) : (Ω, F ) → (R, B (R)) is measurable, for all ¯ . (Recall the convention 0.∞ = 0). a∈R 5. Explain why the map f + g may not be well defined. 6. Suppose that f ≥ 0 and g ≥ 0, i.e. f (Ω) ⊆ [0, +∞] and also g (Ω) ⊆ [0, +∞]. Show that {f + g < c} = {f < c − g }, for all ¯ ¯ c ∈ R. Show that f + g : (Ω, F ) → (R, B (R)) is measurable. Tutorial 4: Measurability 21 ¯ ¯ 7. Show that f + g : (Ω, F ) → (R, B (R)) is measurable, in the case when f and g take values in R. ¯ ¯ 8. Show that 1/f : (Ω, F ) → (R, B (R)) is measurable, in the case when f (Ω) ⊆ R \ {0}. ¯ ¯ 9. Suppose that f is R-valued. Show that f defined by f (ω ) = ¯(ω ) = 1 if f (ω ) = 0, is measurable with f (ω) if f (ω) = 0 and f ¯ respect to F and B (R). ¯ 10. Suppose f and g take values in R. Let f be defined as in 9. Show that for all c ∈ R, the set {f g < c} can be expressed as: ¯ ¯ ({f > 0}∩{g < c/f }) ({f < 0}∩{g > c/f }) ({f = 0}∩{f < c}) ¯ ¯ 11. Show that f g : (Ω, F ) → (R, B (R)) is measurable, in the case when f and g take values in R. Tutorial 4: Measurability 22 ¯ ¯ Exercise 20.Let f, g : (Ω, F ) → (R, B (R)) be two measurable maps, ¯¯ where (Ω, F ) is a measurable space. Let f , g , be defined by: ¯ f (ω ) = f (ω ) 1 if if f (ω) ∈ {−∞, +∞} f (ω) ∈ {−∞, +∞} g (ω ) being defined in a similar way. Consider the partitions of Ω, ¯ Ω = A1 A2 A3 A4 A5 and Ω = B1 B2 B3 B4 B5 , where A1 = {f ∈]0, +∞[}, A2 = {f ∈] − ∞, 0[}, A3 = {f = 0}, A4 = {f = −∞}, A5 = {f = +∞} and B1 , B2 , B3 , B4 , B5 being defined in a similar way with g . Recall the conventions 0 × (+∞) = 0, (−∞) × (+∞) = (−∞), etc. . . ¯ ¯ 1. Show that f and g are measurable with respect to F and B (R). ¯ 2. Show that all Ai ’s and Bj ’s are elements of F . ¯ 3. Show that for all B ∈ B (R): 5 (Ai ∩ Bj ∩ {f g ∈ B }) {f g ∈ B } = i,j =1 Tutorial 4: Measurability 23 ¯¯ 4. Show that Ai ∩ Bj ∩ {f g ∈ B } = Ai ∩ Bj ∩ {f g ∈ B }, in the case when 1 ≤ i ≤ 3 and 1 ≤ j ≤ 3. 5. Show that Ai ∩ Bj ∩ {f g ∈ B } is either equal to ∅ or Ai ∩ Bj , in the case when i ≥ 4 or j ≥ 4. ¯ ¯ 6. Show that f g : (Ω, F ) → (R, B (R)) is measurable. Definition 37 Let (Ω, T ) be a topological space, and A ⊆ Ω. We ¯ call closure of A in Ω, denoted A, the set defined by: ¯ A = {x ∈ Ω : x ∈ U ∈ T ⇒ U ∩ A = ∅} ¯ Exercise 21. Let (E, T ) be a topological space, and A ⊆ E . Let A be the closure of A. ¯ ¯ 1. Show that A ⊆ A and that A is closed. ¯ 2. Show that if B is closed and A ⊆ B , then A ⊆ B . Tutorial 4: Measurability 24 ¯ 3. Show that A is the smallest closed set in E containing A. ¯ 4. Show that A is closed if and only if A = A. 5. Show that if (E, T ) is metrizable, then: ¯ A = {x ∈ E : ∀ > 0 , B (x, ) ∩ A = ∅} d where B (x, ) is relative to any metric d such that TE = T . Exercise 22. Let (E, d) be a metric space. Let A ⊆ E . For all x ∈ E , we define: d(x, A) = inf {d(x, y ) : y ∈ A} = ΦA (x) where it is understood that inf ∅ = +∞. ¯ 1. Show that for all x ∈ E , d(x, A) = d(x, A). ¯ 2. Show that d(x, A) = 0, if and only if x ∈ A. 3. Show that for all x, y ∈ E , d(x, A) ≤ d(x, y ) + d(y, A). Tutorial 4: Measurability 25 4. Show that if A = ∅, |d(x, A) − d(y, A)| ≤ d(x, y ). d ¯¯ 5. Show that ΦA : (E, TE ) → (R, TR ) is continuous. 6. Show that if A is closed, then A = Φ−1 ({0}) A Exercise 23.Let (Ω, F ) be a measurable space. Let (fn )n≥1 be a sequence of measurable maps fn : (Ω, F ) → (E, B (E )), where (E, d) is a metric space. We assume that for all ω ∈ Ω, the sequence (fn (ω ))n≥1 converges to some f (ω ) ∈ E . 1. Explain why lim inf fn and lim sup fn may not be defined in an arbitrary metric space E . 2. Show that f : (Ω, F ) → (E, B (E )) is measurable, if and only if f −1 (A) ∈ F for all closed subsets A of E . 3. Show that for all A closed in E , f −1 (A) = (ΦA ◦ f )−1 ({0}), ¯ where the map ΦA : E → R is defined as in exercise (22). Tutorial 4: Measurability 26 ¯ ¯ 4. Show that ΦA ◦ fn : (Ω, F ) → (R, B (R)) is measurable. 5. Show that f : (Ω, F ) → (E, B (E )) is measurable. Theorem 17 Let (Ω, F ) be a measurable space. Let (fn )n≥1 be a sequence of measurable maps fn : (Ω, F ) → (E, B (E )), where (E, d) is a metric space. Then, if the limit f = lim fn exists on Ω, the map f : (Ω, F ) → (E, B (E )) is itself measurable. Definition 38 The usual topology on C, the set of complex numbers, is defined as the metric topology associated with d(z, z ) = |z −z |. Exercise 24. Let f : (Ω, F ) → (C, B (C)) be a measurable map, where (Ω, F ) is a measurable space. Let u = Re(f ) and v = Im(f ). ¯ ¯ Show that u, v, |f | : (Ω, F ) → (R, B (R)) are all measurable. Exercise 25. Define the subset of the power set P (C): C = {]a, b[×]c, d[ , a, b, c, d ∈ R} Tutorial 4: Measurability 27 where it is understood that:a, b[×]c, d[ = {z = x + iy ∈ C , (x, y ) ∈]a, b[×]c, d[} 1. Show that any element of C is open in C. 2. Show that σ (C ) ⊆ B (C). 3. Let z = x + iy ∈ C. Show that if |x| < η and |y | < η then we √ have |z | < 2η . 4. Let U be open in C. Show that for all z ∈ U , there are rational numbers az , bz , cz , dz such that z ∈]az , bz [×]cz , dz [⊆ U . ∞ 5. Show that U can be written as U = ∪+=1 An where An ∈ C . n 6. Show that σ (C ) = B (C). 7. Let (Ω, F ) be a measurable space, and u, v : (Ω, F ) → (R, B (R)) be two measurable maps. Show that u + iv : (Ω, F ) → (C, B (C)) is measurable. Tutorial 5: Lebesgue Integration 1 5. Lebesgue Integration In the following, (Ω, F , µ) is a measure space. Definition 39 Let A ⊆ Ω. We call characteristic function of A, the map 1A : Ω → R, defined by: ∀ω ∈ Ω , 1A (ω ) = 1 0 if if ω∈A ω∈A ¯ ¯ Exercise 1. Given A ⊆ Ω, show that 1A : (Ω, F ) → (R, B (R)) is measurable if and only if A ∈ F . Definition 40 Let (Ω, F ) be a measurable space. We say that a map s : Ω → R+ is a simple function on (Ω, F ), if and only if s is of the form : n αi 1Ai s= i=1 where n ≥ 1, αi ∈ R+ and Ai ∈ F , for all i = 1, . . . , n. Tutorial 5: Lebesgue Integration 2 Exercise 2. Show that s : (Ω, F ) → (R+ , B (R+ )) is measurable, whenever s is a simple function on (Ω, F ). Exercise 3. Let s be a simple function on (Ω, F ) with representation n n s= i=1 αi 1Ai . Consider the map φ : Ω → {0, 1} defined by φ(ω ) = (1A1 (ω ), . . . , 1An (ω )). For each y ∈ s(Ω), pick one ω y ∈ Ω such that y = s(ω y ). Consider the map ψ : s(Ω) → {0, 1}n defined by ψ (y ) = φ(ω y ). 1. Show that ψ is injective, and that s(Ω) is a finite subset of R+ . 2. Show that s = α∈s(Ω) α1{s=α} 3. Show that any simple function s can be represented as: n αi 1Ai s= i=1 where n ≥ 1, αi ∈ R+ , Ai ∈ F and Ω = A1 ... An . Tutorial 5: Lebesgue Integration 3 Definition 41 Let (Ω, F ) be a measurable space, and s be a simple function on (Ω, F ). We call partition of the simple function s, any representation of the form: n αi 1Ai s= i=1 where n ≥ 1, αi ∈ R+ , Ai ∈ F and Ω = A1 ... An . Exercise 4. Let s be a simple function on (Ω, F ) with two partitions: n i=1 1. Show that s = i,j m αi 1Ai = s= βj 1Bj j =1 αi 1Ai ∩Bj is a partition of s. 2. Recall the convention 0 × (+∞) = 0 and α × (+∞) = +∞ if α > 0. For all a1 , . . . , ap in [0, +∞], p ≥ 1 and x ∈ [0, +∞], prove the distributive property: x(a1 +. . .+ap ) = xa1 +. . .+xap . Tutorial 5: Lebesgue Integration 3. Show that n i=1 4 m j =1 αi µ(Ai ) = βj µ(Bj ). 4. Explain why the following definition is legitimate. Definition 42 Let (Ω, F , µ) be a measure space, and s be a simple function on (Ω, F ). We define the integral of s with respect to µ, as the sum, denoted I µ (s), defined by: n αi µ(Ai ) ∈ [0, +∞] I µ (s) = i=1 where s = n i=1 αi 1Ai is any partition of s. Exercise 5. Let s, t be two simple functions on (Ω, F ) with partitions s = n αi 1Ai and t = m βj 1Bj . Let α ∈ R+ . i=1 j =1 1. Show that s + t is a simple function on (Ω, F ) with partition: n m (αi + βj )1Ai ∩Bj s+t= i=1 j =1 Tutorial 5: Lebesgue Integration 5 2. Show that I µ (s + t) = I µ (s) + I µ (t). 3. Show that αs is a simple function on (Ω, F ). 4. Show that I µ (αs) = αI µ (s). 5. Why is the notation I µ (αs) meaningless if α = +∞ or α < 0. 6. Show that if s ≤ t then I µ (s) ≤ I µ (t). Exercise 6. Let f : (Ω, F ) → [0, +∞] be a non-negative and measurable map. For all n ≥ 1, we define: n2n −1 sn = k=0 k 1k k+1 + n1{n≤f } 2n { 2n ≤f < 2n } 1. Show that sn is a simple function on (Ω, F ), for all n ≥ 1. 2. Show that equation (1) is a partition sn , for all n ≥ 1. 3. Show that sn ≤ sn+1 ≤ f , for all n ≥ 1. (1) Tutorial 5: Lebesgue Integration 6 4. Show that sn ↑ f as n → +∞1 . Theorem 18 Let f : (Ω, F ) → [0, +∞] be a non-negative and measurable map, where (Ω, F ) is a measurable space. There exists a sequence (sn )n≥1 of simple functions on (Ω, F ) such that sn ↑ f . Definition 43 Let f : (Ω, F ) → [0, +∞] be a non-negative and measurable map, where (Ω, F , µ) is a measure space. We define the lebesgue integral of f with respect to µ, denoted f dµ, as: f dµ = sup{I µ (s) : s simple function on (Ω, F ) , s ≤ f } where, given any simple function s on (Ω, F ), I µ (s) denotes its integral with respect to µ. 1 i.e. for all ω ∈ Ω, the sequence (s (ω )) n n≥1 is non-decreasing and converges to f (ω ) ∈ [0, +∞]. Tutorial 5: Lebesgue Integration 7 Exercise 7. Let f : (Ω, F ) → [0, +∞] be a non-negative and measurable map. 1. Show that f dµ ∈ [0, +∞]. 2. Show that f dµ = I µ (f ), whenever f is a simple function. 3. Show that g dµ ≤ f dµ, whenever g : (Ω, F ) → [0, +∞] is non-negative and measurable map with g ≤ f . 4. Show that (cf )dµ = c f dµ, if 0 < c < +∞. Explain why both integrals are well defined. Is the equality still true for c = 0. 5. For n ≥ 1, put An = {f > 1/n}, and sn = (1/n)1An . Show that sn is a simple function on (Ω, F ) with sn ≤ f . Show that An ↑ {f > 0}. 6. Show that f dµ = 0 ⇒ µ({f > 0}) = 0. Tutorial 5: Lebesgue Integration 8 7. Show that if s is a simple function on (Ω, F ) with s ≤ f , then µ({f > 0}) = 0 implies I µ (s) = 0. 8. Show that f dµ = 0 ⇔ µ({f > 0}) = 0. 9. Show that (+∞)f dµ = (+∞) f dµ. Explain why both integrals are well defined. 10. Show that (+∞)1{f =+∞} ≤ f and: (+∞)1{f =+∞} dµ = (+∞)µ({f = +∞}) 11. Show that f dµ < +∞ ⇒ µ({f = +∞}) = 0. 12. Suppose that µ(Ω) = +∞ and take f = 1. Show that the converse of the previous implication is not true. Tutorial 5: Lebesgue Integration 9 Exercise 8. Let s be a simple function on (Ω, F ). let A ∈ F . 1. Show that s1A is a simple function on (Ω, F ). 2. Show that for any partition s = n i=1 αi 1Ai of s, we have: n αi µ(Ai ∩ A) I µ (s1A ) = i=1 3. Let ν : F → [0, +∞] be defined by ν (A) = I µ (s1A ). Show that ν is a measure on F . 4. Suppose An ∈ F , An ↑ A. Show that I µ (s1An ) ↑ I µ (s1A ). Exercise 9. Let (fn )n≥1 be a sequence of non-negative and measurable maps fn : (Ω, F ) → [0, +∞], such that fn ↑ f . 1. Recall what the notation fn ↑ f means. ¯ ¯ 2. Explain why f : (Ω, F ) → (R, B (R)) is measurable. Tutorial 5: Lebesgue Integration 10 3. Let α = supn≥1 fn dµ. Show that 4. Show that α ≤ fn dµ ↑ α. f dµ. 5. Let s be any simple function on (Ω, F ) such that s ≤ f . Let c ∈]0, 1[. For n ≥ 1, define An = {cs ≤ fn }. Show that An ∈ F and An ↑ Ω. 6. Show that cI µ (s1An ) ≤ fn dµ, for all n ≥ 1. 7. Show that cI µ (s) ≤ α. 8. Show that I µ (s) ≤ α. 9. Show that f dµ ≤ α. 10. Conclude that fn dµ ↑ f dµ. Theorem 19 (Monotone Convergence) Let (Ω, F , µ) be a measure space. Let (fn )n≥1 be a sequence of non-negative and measurable maps fn : (Ω, F ) → [0, +∞] such that fn ↑ f . Then fn dµ ↑ f dµ. Tutorial 5: Lebesgue Integration 11 Exercise 10. Let f, g : (Ω, F ) → [0, +∞] be two non-negative and measurable maps. Let a, b ∈ [0, +∞]. 1. Show that if (fn )n≥1 and (gn )n≥1 are two sequences of nonnegative and measurable maps such that fn ↑ f and gn ↑ g , then fn + gn ↑ f + g . 2. Show that (f + g )dµ = f dµ + g dµ. 3. Show that (af + bg )dµ = a f dµ + b g dµ. Exercise 11. Let (fn )n≥1 be a sequence of non-negative and mea+∞ surable maps fn : (Ω, F ) → [0, +∞]. Define f = n=1 fn . 1. Explain why f : (Ω, F ) → [0, +∞] is well defined, non-negative and measurable. 2. Show that f dµ = +∞ n=1 fn dµ. Tutorial 5: Lebesgue Integration 12 Definition 44 Let (Ω, F , µ) be a measure space and let P (ω) be a property depending on ω ∈ Ω. We say that the property P (ω ) holds µ-almost surely, and we write P (ω) µ-a.s., if and only if: ∃N ∈ F , µ(N ) = 0 , ∀ω ∈ N c , P (ω) holds Exercise 12. Let P (ω) be a property depending on ω ∈ Ω, such that {ω ∈ Ω : P (ω) holds} is an element of the σ -algebra F . 1. Show that P (ω) , µ-a.s. ⇔ µ({ω ∈ Ω : P (ω) holds}c ) = 0. 2. Explain why in general, the right-hand side of this equivalence cannot be used to defined µ-almost sure properties. Exercise 13. Let (Ω, F , µ) be a measure space and (An )n≥1 be a +∞ ∞ sequence of elements of F . Show that µ(∪+=1 An ) ≤ n=1 µ(An ). n Exercise 14. Let (fn )n≥1 be a sequence of maps fn : Ω → [0, +∞]. 1. Translate formally the statement fn ↑ f µ-a.s. Tutorial 5: Lebesgue Integration 13 2. Translate formally fn → f µ-a.s. and ∀n, (fn ≤ fn+1 µ-a.s.) 3. Show that the statements 1. and 2. are equivalent. Exercise 15. Suppose that f, g : (Ω, F ) → [0, +∞] are non-negative and measurable with f = g µ-a.s.. Let N ∈ F , µ(N ) = 0 such that f = g on N c . Explain why f dµ = (f 1N )dµ + (f 1N c )dµ, all integrals being well defined. Show that f dµ = g dµ. Exercise 16. Suppose (fn )n≥1 is a sequence of non-negative and measurable maps such that fn ↑ f µ-a.s.. Let N ∈ F , µ(N ) = 0, such ¯ ¯ that fn ↑ f on N c . Define fn = fn 1N c and f = f 1N c . ¯ ¯ 1. Explain why f and the fn ’s are non-negative and measurable. ¯ ¯ 2. Show that fn ↑ f . 3. Show that fn dµ ↑ f dµ. Tutorial 5: Lebesgue Integration 14 Exercise 17. Let (fn )n≥1 be a sequence of non-negative and measurable maps fn : (Ω, F ) → [0, +∞]. For n ≥ 1, we define gn = inf k≥n fk . 1. Explain why the gn ’s are non-negative and measurable. 2. Show that gn ↑ lim inf fn . 3. Show that gn dµ ≤ fn dµ, for all n ≥ 1. ¯ 4. Show that if (un )n≥1 and (vn )n≥1 are two sequences in R with un ≤ vn for all n ≥ 1, then lim inf un ≤ lim inf vn . 5. Show that (lim inf fn )dµ ≤ lim inf integrals are well defined. fn dµ, and recall why all Theorem 20 (Fatou Lemma) Let (Ω, F , µ) be a measure space, and let (fn )n≥1 be a sequence of non-negative and measurable maps fn : (Ω, F ) → [0, +∞]. Then: (lim inf fn )dµ ≤ lim inf n→+∞ n→+∞ fn dµ Tutorial 5: Lebesgue Integration 15 Exercise 18. Let f : (Ω, F ) → [0, +∞] be a non-negative and measurable map. Let A ∈ F . 1. Recall what is meant by the induced measure space (A, F|A , µ|A). Why is it important to have A ∈ F . Show that the restriction of f to A, f|A : (A, F|A ) → [0, +∞] is measurable. 2. We define the map µA : F → [0, +∞] by µA (E ) = µ(A ∩ E ), for all E ∈ F . Show that (Ω, F , µA ) is a measure space. 3. Consider the equalities: (f 1A )dµ = f dµA = (f|A )dµ|A (2) For each of the above integrals, what is the underlying measure space on which the integral is considered. What is the map being integrated. Explain why each integral is well defined. 4. Show that in order to prove (2), it is sufficient to consider the case when f is a simple function on (Ω, F ). Tutorial 5: Lebesgue Integration 16 5. Show that in order to prove (2), it is sufficient to consider the case when f is of the form f = 1B , for some B ∈ F . 6. Show that (2) is indeed true. Definition 45 Let f : (Ω, F ) → [0, +∞] be a non-negative and measurable map, where (Ω, F , µ) is a measure space. let A ∈ F . We call partial lebesgue integral of f with respect to µ over A, the integral denoted A f dµ, defined as: f dµ = (f 1A )dµ = f dµA = (f|A )dµ|A A where µA is the measure on (Ω, F ), µA = µ(A ∩ •), f|A is the restriction of f to A and µ|A is the restriction of µ to F|A , the trace of F on A. Tutorial 5: Lebesgue Integration 17 Exercise 19. Let f, g : (Ω, F ) → [0, +∞] be two non-negative and measurable maps. Let ν : F → [0, +∞] be defined by ν (A) = A f dµ, for all A ∈ F . 1. Show that ν is a measure on F . 2. Show that: g dν = g f dµ Theorem 21 Let f : (Ω, F ) → [0, +∞] be a non-negative and measurable map, where (Ω, F , µ) is a measure space. Let ν : F → [0, +∞] be defined by ν (A) = A f dµ, for all A ∈ F . Then, ν is a measure on F , and for all g : (Ω, F ) → [0, +∞] non-negative and measurable, we have: g dν = g f dµ Tutorial 5: Lebesgue Integration 18 Definition 46 The L1 -spaces on a measure space (Ω, F , µ), are: L1 (Ω, F , µ)= f : (Ω, F ) → (R, B (R)) measurable, R |f |dµ < +∞ L1 (Ω, F , µ)= f : (Ω, F ) → (C, B (C)) measurable, C |f |dµ < +∞ Exercise 20. Let f : (Ω, F ) → (C, B (C)) be a measurable map. 1. Explain why the integral |f |dµ makes sense. 2. Show that f : (Ω, F ) → (R, B (R)) is measurable, if f (Ω) ⊆ R. 3. Show that L1 (Ω, F , µ) ⊆ L1 (Ω, F , µ). R C 4. Show that L1 (Ω, F , µ) = {f ∈ L1 (Ω, F , µ) , f (Ω) ⊆ R} R C 5. Show that L1 (Ω, F , µ) is closed under R-linear combinations. R 6. Show that L1 (Ω, F , µ) is closed under C-linear combinations. C Tutorial 5: Lebesgue Integration 19 Definition 47 Let u : Ω → R be a real-valued function defined on a set Ω. We call positive part and negative part of u the maps u+ and u− respectively, defined as u+ = max(u, 0) and u− = max(−u, 0). Exercise 21. Let f ∈ L1 (Ω, F , µ). Let u = Re(f ) and v = Im(f ). C 1. Show that u = u+ − u− , v = v + − v − , f = u+ − u− + i(v + − v − ). 2. Show that |u| = u+ + u− , |v | = v + + v − 3. Show that u+ , u− , v + , v − , |f |, u, v, |u|, |v | all lie in L1 (Ω, F , µ). R 4. Explain why the integrals all well defined. u+ dµ, u− dµ, v + dµ, v − dµ are 5. We define the integral of f with respect to µ, denoted f dµ, as f dµ = u+ dµ − u− dµ + i v + dµ − v − dµ . Explain why f dµ is a well defined complex number. Tutorial 5: Lebesgue Integration 20 6. In the case when f (Ω) ⊆ C ∩ [0, +∞] = R+ , explain why this new definition of the integral of f with respect to µ is consistent with the one already known (43) for non-negative and measurable maps. 7. Show that f dµ = udµ + i v dµ and explain why all integrals involved are well defined. Definition 48 Let f = u + iv ∈ L1 (Ω, F , µ) where (Ω, F , µ) is a C measure space. We define the lebesgue integral of f with respect to µ, denoted f dµ, as: f dµ = u+ dµ − u− dµ + i v + dµ − v − dµ Exercise 22. Let f = u + iv ∈ L1 (Ω, F , µ) and A ∈ F . C 1. Show that f 1A ∈ L1 (Ω, F , µ). C Tutorial 5: Lebesgue Integration 21 2. Show that f ∈ L1 (Ω, F , µA ). C 3. Show that f|A ∈ L1 (A, F|A , µ|A ) C 4. Show that (f 1A )dµ = 5. Show that 4. is: A f dµA = u+ dµ − A f|A dµ|A . u− dµ + i A v + dµ − A v − dµ . Definition 49 Let f ∈ L1 (Ω, F , µ) , where (Ω, F , µ) is a measure C space. let A ∈ F . We call partial lebesgue integral of f with respect to µ over A, the integral denoted A f dµ, defined as: f dµ = A (f 1A )dµ = f dµA = (f|A )dµ|A where µA is the measure on (Ω, F ), µA = µ(A ∩ •), f|A is the restriction of f to A and µ|A is the restriction of µ to F|A , the trace of F on A. Tutorial 5: Lebesgue Integration 22 Exercise 23. Let f, g ∈ L1 (Ω, F , µ) and let h = f + g R 1. Show that: h+ dµ + 2. Show that f − dµ + hdµ = g − dµ = f dµ + h− dµ + f + dµ + g dµ. 3. Show that (−f )dµ = − f dµ 4. Show that if α ∈ R then (αf )dµ = α f dµ. 5. Show that if f ≤ g then f dµ ≤ g dµ 6. Show the following theorem. Theorem 22 For all f, g ∈ L1 (Ω, F , µ) and α ∈ C, we have: C (αf + g )dµ = α f dµ + g dµ g + dµ Tutorial 5: Lebesgue Integration 23 Exercise 24. Let f, g be two maps, and (fn )n≥1 be a sequence of measurable maps fn : (Ω, F ) → (C, B (C)), such that: (i) ∀ω ∈ Ω , lim fn (ω ) = f (ω) in C n→+∞ (ii) ∀n ≥ 1 , |fn | ≤ g (iii) g ∈ L1 (Ω, F , µ) R ¯ Let (un )n≥1 be an arbitrary sequence in R. 1. Show that f ∈ L1 (Ω, F , µ) and fn ∈ L1 (Ω, F , µ) for all n ≥ 1. C C 2. For n ≥ 1, define hn = 2g − |fn − f |. Explain why Fatou lemma (20) can be applied to the sequence (hn )n≥1 . 3. Show that lim inf(−un ) = − lim sup un . 4. Show that if α ∈ R, then lim inf(α + un ) = α + lim inf un . 5. Show that un → 0 as n → +∞ if and only if lim sup |un | = 0. 6. Show that (2g )dµ ≤ (2g )dµ − lim sup |fn − f |dµ Tutorial 5: Lebesgue Integration 24 7. Show that lim sup |fn − f |dµ = 0. 8. Conclude that |fn − f |dµ → 0 as n → +∞. Theorem 23 (Dominated Convergence) Let (fn )n≥1 be a sequence of measurable maps fn : (Ω, F ) → (C, B (C)) such that fn → f in C2 . Suppose that there exists some g ∈ L1 (Ω, F , µ) such that R |fn | ≤ g for all n ≥ 1. Then f, fn ∈ L1 (Ω, F , µ) for all n ≥ 1, and: C lim n→+∞ |fn − f |dµ = 0 Exercise 25. Let f ∈ L1 (Ω, F , µ) and put z = f dµ. Let α ∈ C, C be such that |α| = 1 and αz = |z |. Put u = Re(αf ). 1. Show that u ∈ L1 (Ω, F , µ) R 2. Show that u ≤ |f | 2 i.e. for all ω ∈ Ω, the sequence (fn (ω ))n≥1 converges to f (ω ) ∈ C Tutorial 5: Lebesgue Integration 25 3. Show that | f dµ| = (αf )dµ. 4. Show that (αf )dµ = udµ. 5. Prove the following theorem. Theorem 24 Let f ∈ L1 (Ω, F , µ) where (Ω, F , µ) is a measure C space. We have: f dµ ≤ |f |dµ Tutorial 6: Product Spaces 1 6. Product Spaces In the following, I is a non-empty set. Definition 50 Let (Ωi )i∈I be a family of sets, indexed by a nonempty set I . We call cartesian product of the family (Ωi )i∈I the set, denoted Πi∈I Ωi , and defined by: Ωi = {ω : I → ∪i∈I Ωi , ω (i) ∈ Ωi , ∀i ∈ I } i∈I In other words, Πi∈I Ωi is the set of all maps ω defined on I , with values in ∪i∈I Ωi , such that ω (i) ∈ Ωi for all i ∈ I . Theorem 25 (Axiom of choice) Let (Ωi )i∈I be a family of sets, indexed by a non-empty set I . Then, Πi∈I Ωi is non-empty, if and only if Ωi is non-empty for all i ∈ I 1 . 1 When I is finite, this theorem is traditionally derived from other axioms. Tutorial 6: Product Spaces 2 Exercise 1. 1. Let Ω be a set and suppose that Ωi = Ω, ∀i ∈ I . We use the notation ΩI instead of Πi∈I Ωi . Show that ΩI is the set of all maps ω : I → Ω. + ¯R 2. What are the sets RR , RN , [0, 1]N , R ? ∞ 3. Suppose I = N∗ . We sometimes use the notation Π+=1 Ωn inn stead of Πn∈N∗ Ωn . Let S be the set of all sequences (xn )n≥1 such that xn ∈ Ωn for all n ≥ 1. Is S the same thing as the ∞ product Π+=1 Ωn ? n 4. Suppose I = Nn = {1, . . . , n}, n ≥ 1. We use the notation Ω1 × . . . × Ωn instead of Πi∈{1,...,n} Ωi . For ω ∈ Ω1 × . . . × Ωn , it is customary to write (ω 1 , . . . , ωn ) instead of ω , where we have ωi = ω(i). What is your guess for the definition of sets such as ¯n Rn , R , Qn , Cn . 5. Let E, F, G be three sets. Define E × F × G. Tutorial 6: Product Spaces 3 Definition 51 Let I be a non-empty set. We say that a family of sets (Iλ )λ∈Λ , where Λ = ∅, is a partition of I , if and only if: (i) (ii) (iii) ∀λ ∈ Λ , Iλ = ∅ ∀λ, λ ∈ Λ , λ = λ ⇒ Iλ ∩ Iλ = ∅ I = ∪λ∈Λ Iλ Exercise 2. Let (Ωi )i∈I be a family of sets indexed by I , and (Iλ )λ∈Λ be a partition of the set I . 1. For each λ ∈ Λ, recall the definition of Πi∈Iλ Ωi . 2. Recall the definition of Πλ∈Λ (Πi∈Iλ Ωi ). 3. Define a natural bijection Φ : Πi∈I Ωi → Πλ∈Λ (Πi∈Iλ Ωi ). 4. Define a natural bijection ψ : Rp × Rn → Rp+n , for all n, p ≥ 1. Tutorial 6: Product Spaces 4 Definition 52 Let (Ωi )i∈I be a family of sets, indexed by a nonempty set I . For all i ∈ I , let Ei be a set of subsets of Ωi . We define a rectangle of the family (Ei )i∈I , as any subset A of Πi∈I Ωi , of the form A = Πi∈I Ai where Ai ∈ Ei ∪ {Ωi } for all i ∈ I , and such that Ai = Ωi except for a finite number of indices i ∈ I . Consequently, the set of all rectangles, denoted i∈I Ei , is defined as: Ei = i∈I Ai : Ai ∈ Ei ∪ {Ωi } , Ai = Ωi for finitely many i ∈ I i∈I Exercise 3. (Ωi )i∈I and (Ei )i∈I being as above: 1. Show that if I = Nn and Ωi ∈ Ei for all i = 1, . . . , n, then E1 . . . En = {A1 × . . . × An : Ai ∈ Ei , ∀i ∈ I }. 2. Let A be a rectangle. Show that there exists a finite subset J of I such that: A = {ω ∈ Πi∈I Ωi : ω(j ) ∈ Aj , ∀j ∈ J } for some Aj ’s such that Aj ∈ Ej , for all j ∈ J . Tutorial 6: Product Spaces 5 Definition 53 Let (Ωi , Fi )i∈I be a family of measurable spaces, indexed by a non-empty set I . We call measurable rectangle , any rectangle of the family (Fi )i∈I . The set of all measurable rectangles is given by 2 : Fi = i∈I Ai : Ai ∈ Fi , Ai = Ωi for finitely many i ∈ I i∈I Definition 54 Let (Ωi , Fi )i∈I be a family of measurable spaces, indexed by a non-empty set I . We define the product σ -algebra of (Fi )i∈I , as the σ -algebra on Πi∈I Ωi , denoted ⊗i∈I Fi , and generated by all measurable rectangles, i.e. Fi = σ i∈I 2 Note that Ωi ∈ Fi for all i ∈ I . Fi i∈I Tutorial 6: Product Spaces 6 Exercise 4. 1. Suppose I = Nn . Show that F1 ⊗ . . . ⊗ Fn is generated by all sets of the form A1 × . . . × An , where Ai ∈ Fi for all i = 1, . . . , n. 2. Show that B (R) ⊗ B (R) ⊗ B (R) is generated by sets of the form A × B × C where A, B, C ∈ B (R). 3. Show that if (Ω, F ) is a measurable space, B (R+ ) ⊗ F is the σ -algebra on R+ × Ω generated by sets of the form B × F where B ∈ B (R+ ) and F ∈ F . Exercise 5. Let (Ωi )i∈I be a family of non-empty sets and Ei be a subset of the power set P (Ωi ) for all i ∈ I . 1. Give a generator of the σ -algebra ⊗i∈I σ (Ei ) on Πi∈I Ωi . 2. Show that: Ei σ i∈I σ (Ei ) ⊆ i∈I Tutorial 6: Product Spaces 7 3. Let A be a rectangle of the family (σ (Ei ))i∈I . Show that if A is not empty, then the representation A = Πi∈I Ai with Ai ∈ σ (Ei ) is unique. Define JA = {i ∈ I : Ai = Ωi }. Explain why JA is a well-defined finite subset of I . 4. If A ∈ i∈I σ (Ei ), Show that if A = ∅, or A = ∅ and JA = ∅, then A ∈ σ ( i∈I Ei ). Exercise 6. Everything being as before, Let n ≥ 0. We assume that the following induction hypothesis has been proved: σ (Ei ), A = ∅, cardJA = n ⇒ A ∈ σ A∈ i∈I Ei i∈I We assume that A is a non empty measurable rectangle of (σ (Ei ))i∈I with cardJA = n + 1. Let JA = {i1 , . . . , in+1 } be an extension of JA . For all B ⊆ Ωi1 , we define: ¯ Ai AB = i∈I Tutorial 6: Product Spaces 8 ¯ ¯ where each Ai is equal to Ai except Ai1 = B . We define the set: Γ= B ⊆ Ωi1 : AB ∈ σ Ei i∈I 1. Show that AΩi1 = ∅, cardJAΩi1 = n and that AΩi1 ∈ i∈I σ (Ei ). 2. Show that Ωi1 ∈ Γ. 3. Show that for all B ⊆ Ωi1 , we have AΩi1 \B = AΩi1 \ AB . 4. Show that B ∈ Γ ⇒ Ωi1 \ B ∈ Γ. 5. Let Bn ⊆ Ωi1 , n ≥ 1. Show that A∪Bn = ∪n≥1 ABn . 6. Show that Γ is a σ -algebra on Ωi1 . ¯ 7. Let B ∈ Ei1 , and for i ∈ I define Bi = Ωi for all i’s except ¯ ¯ i1 = B . Show that AB = AΩi1 ∩ (Πi∈I Bi ). B 8. Show that σ (Ei1 ) ⊆ Γ. Tutorial 6: Product Spaces 9 9. Show that A = AAi1 and A ∈ σ ( 10. Show that i∈I σ (Ei ) 11. Show that σ ( i∈I Ei ) ⊆ σ( i∈I Ei ). i∈I Ei ). = ⊗i∈I σ (Ei ). Theorem 26 Let (Ωi )i∈I be a family of non-empty sets indexed by a non-empty set I . For all i ∈ I , let Ei be a set of subsets of Ωi . Then, the product σ -algebra ⊗i∈I σ (Ei ) on the cartesian product Πi∈I Ωi is generated by the rectangles of (Ei )i∈I , i.e. : σ (Ei ) = σ i∈I Ei i∈I Exercise 7. Let TR denote the usual topology in R. Let n ≥ 1. 1. Show that TR ... TR = {A1 × . . . × An : Ai ∈ TR }. 2. Show that B (R) ⊗ . . . ⊗ B (R) = σ (TR ... TR ). Tutorial 6: Product Spaces 10 3. Define C2 = {]a1 , b1 ] × . . . ×]an , bn ] : ai , bi ∈ R}. Show that C2 ⊆ S . . . S , where S = {]a, b] : a, b ∈ R}, but that the inclusion is strict. 4. Show that S ... S ⊆ σ (C2 ). 5. Show that B (R) ⊗ . . . ⊗ B (R) = σ (C2 ). Exercise 8. Let Ω and Ω be two non-empty sets. Let A be a subset of Ω such that ∅ = A = Ω. Let E = {A} ⊆ P (Ω) and E = ∅ ⊆ P (Ω ). 1. Show that σ (E ) = {∅, A, Ac , Ω}. 2. Show that σ (E ) = {∅, Ω }. 3. Define C = {E × F , E ∈ E , F ∈ E } and show that C = ∅. 4. Show that E E = {A × Ω , Ω × Ω }. 5. Show that σ (E ) ⊗ σ (E ) = {∅, A × Ω , Ac × Ω , Ω × Ω }. 6. Conclude that σ (E ) ⊗ σ (E ) = σ (C ) = {∅, Ω × Ω }. Tutorial 6: Product Spaces 11 Exercise 9. Let n ≥ 1 and p ≥ 1 be two positive integers. 1. Define F = B (R) ⊗ . . . ⊗ B (R), and G = B (R) ⊗ . . . ⊗ B (R). n p Explain why F ⊗ G can be viewed as a σ -algebra on Rn+p . 2. Show that F ⊗G is generated by sets of the form A1 × . . . × An+p where Ai ∈ B (R), i = 1, . . . , n + p. 3. Show that: B (R) ⊗ . . . ⊗B (R) = (B (R) ⊗ . . . ⊗B (R))⊗(B (R) ⊗ . . . ⊗B (R)) n+p n p Exercise 10. Let (Ωi , Fi )i∈I be a family of measurable spaces. Let (Iλ )λ∈Λ , where Λ = ∅, be a partition of I . Let Ω = Πi∈I Ωi and Ω Ω = Πλ∈Λ (Πi∈Iλ i ). 1. Define a natural bijection between P (Ω) and P (Ω ). Tutorial 6: Product Spaces 12 2. Show that through such bijection, A = Πi∈I Ai ⊆ Ω, where Ai ⊆ Ωi , is identified with A = Πλ∈Λ (Πi∈Iλ Ai ) ⊆ Ω . 3. Show that i∈I Fi = λ∈Λ ( i∈Iλ Fi ). 4. Show that ⊗i∈I Fi = ⊗λ∈Λ (⊗i∈Iλ Fi ). Definition 55 Let Ω be set and A be a set of subsets of Ω. We call topology generated by A, the topology on Ω, denoted T (A), equal to the intersection of all topologies on Ω, which contain A. Exercise 11. Let Ω be a set and A ⊆ P (Ω). 1. Explain why T (A) is indeed a topology on Ω. 2. Show that T (A) is the smallest topology T such that A ⊆ T . 3. Show that the metric topology on a metric space (E, d) is generated by the open balls A = {B (x, ) : x ∈ E, > 0}. Tutorial 6: Product Spaces 13 Definition 56 Let (Ωi , Ti )i∈I be a family of topological spaces, indexed by a non-empty set I . We define the product topology of (Ti )i∈I , as the topology on Πi∈I Ωi , denoted i∈I Ti , and generated by all rectangles of (Ti )i∈I , i.e. Ti = T i∈I Ti i∈I Exercise 12. Let (Ωi , Ti )i∈I be a family of topological spaces. 1. Show that U ∈ i∈I Ti , if and only if: ∀x ∈ U , ∃V ∈ 2. Show that i∈I Ti ⊆ i∈I Ti i∈I Ti . 3. Show that ⊗i∈I B (Ωi ) = σ ( i∈I Ti ). 4. Show that ⊗i∈I B (Ωi ) ⊆ B (Πi∈I Ωi ). , x∈V ⊆U Tutorial 6: Product Spaces 14 Exercise 13. Let n ≥ 1 be a positive integer. For all x, y ∈ Rn , let: n (x, y ) = xi yi i=1 and we put x = (x, x). 1. Show that for all t ∈ R, x + ty 2. From x + ty 2 2 =x 2 + t2 y 2 + 2t(x, y ). ≥ 0 for all t, deduce that |(x, y )| ≤ x . y . 3. Conclude that x + y ≤ x + y . Exercise 14. Let (Ω1 , T1 ), . . . , (Ωn , Tn ), n ≥ 1, be metrizable topological spaces. Let d1 , . . . , dn be metrics on Ω1 , . . . , Ωn , inducing the topologies T1 , . . . , Tn respectively. Let Ω = Ω1 × . . . × Ωn and T be the product topology on Ω. For all x, y ∈ Ω, we define: n (di (xi , yi ))2 d(x, y ) = i=1 Tutorial 6: Product Spaces 15 1. Show that d : Ω × Ω → R+ is a metric on Ω. 2. Show that U ⊆ Ω is open in Ω, if and only if, for all x ∈ U there are open sets U1 , . . . , Un in Ω1 , . . . , Ωn respectively, such that: x ∈ U1 × . . . × Un ⊆ U 3. Let U ∈ T and x ∈ U . Show the existence of > 0 such that: (∀i = 1, . . . , n di (xi , yi ) < ) ⇒ y ∈ U d 4. Show that T ⊆ TΩ . d 5. let U ∈ TΩ and x ∈ U . Show that existence of > 0 such that: x ∈ B (x1 , ) × . . . × B (xn , ) ⊆ U d 6. Show that TΩ ⊆ T . 7. Show that the product topological space (Ω, T ) is metrizable. Tutorial 6: Product Spaces 16 8. For all x, y ∈ Ω, define: n d (x, y ) = di (xi , yi ) i=1 d (x, y ) = max di (xi , yi ) i=1,...,n Show that d , d are metrics on Ω. 9. Show the existence of α , β , α and β > 0, such that we have α d ≤ d ≤ β d and α d ≤ d ≤ β d . 10. Show that d and d also induce the product topology on Ω. Exercise 15. Let (Ωn , Tn )n≥1 be a sequence of metrizable topological spaces. For all n ≥ 1, let dn be a metric on Ωn inducing the topology ∞ Tn . Let Ω = Π+=1 Ωn be the cartesian product and T be the product n Tutorial 6: Product Spaces 17 topology on Ω. For all x, y ∈ Ω, we define: +∞ d(x, y ) = 1 (1 ∧ dn (xn , yn )) 2n n=1 1. Show that for all a, b ∈ R+ , we have 1 ∧ (a + b) ≤ 1 ∧ a + 1 ∧ b. 2. Show that d is a metric on Ω. 3. Show that U ⊆ Ω is open in Ω, if and only if, for all x ∈ U , there is an integer N ≥ 1 and open sets U1 , . . . , UN in Ω1 , . . . , ΩN respectively, such that: +∞ x ∈ U1 × . . . × UN × Ωn ⊆ U n=N +1 4. Show that d(x, y ) < 1/2n ⇒ dn (xn , yn ) ≤ 2n d(x, y ). 5. Show that for all U ∈ T and x ∈ U , there exists > 0 such that d(x, y ) < ⇒ y ∈ U . Tutorial 6: Product Spaces 18 d 6. Show that T ⊆ TΩ . d 7. Let U ∈ TΩ and x ∈ U . Show the existence of > 0 and N ≥ 1, such that: N 1 (1 ∧ dn (xn , yn )) < 2n n=1 ⇒ y∈U d 8. Show that for all U ∈ TΩ and x ∈ U , there is such that: > 0 and N ≥ 1 +∞ x ∈ B (x1 , ) × . . . × B (xN , ) × Ωn ⊆ U n=N +1 d 9. Show that TΩ ⊆ T . 10. Show that the product topological space (Ω, T ) is metrizable. Tutorial 6: Product Spaces 19 Definition 57 Let (Ω, T ) be a topological space. A subset H of T is called a countable base of (Ω, T ), if and only if H is at most countable, and has the property: ∀U ∈ T , ∃H ⊆ H , U = V V ∈H Exercise 16. 1. Show that H = {]r, q [ : r, q ∈ Q} is a countable base of (R, TR ). 2. Show that if (Ω, T ) is a topological space with countable base, and Ω ⊆ Ω, then the induced topological space (Ω , T|Ω ) also has a countable base. 3. Show that [−1, 1] has a countable base. 4. Show that if (Ω, T ) and (S, TS ) are homeomorphic, then (Ω, T ) has a countable base if and only if (S, TS ) has a countable base. ¯¯ 5. Show that (R, TR ) has a countable base. Tutorial 6: Product Spaces 20 Exercise 17. Let (Ωn , Tn )n≥1 be a sequence of topological spaces k with countable base. For n ≥ 1, Let {Vn : k ∈ In } be a countable base of (Ωn , Tn ) where In is a finite or countable set. Let Ω = Π∞ Ωn n=1 be the cartesian product and T be the product topology on Ω. For all p ≥ 1, we define: +∞ Hp = V1k1 × . . . × Vpkp × Ωn : (k1 , . . . , kp ) ∈ I1 × . . . × Ip n=p+1 and we put H = ∪p≥1 Hp . 1. Show that for all p ≥ 1, Hp ⊆ T . 2. Show that H ⊆ T . 3. For all p ≥ 1, show the existence of an injection jp : Hp → Np . 4. Show the existence of a bijection φ2 : N2 → N. 5. For p 1, show the existence of an bijection φp : Np → N. Tutorial 6: Product Spaces 21 6. Show that Hp is at most countable for all p ≥ 1. 7. Show the existence of an injection j : H → N2 . 8. Show that H is a finite or countable set of open sets in Ω. 9. Let U ∈ T and x ∈ U . Show that there is p ≥ 1 and U1 , . . . , Up open sets in Ω1 , . . . , Ωp such that: +∞ x ∈ U1 × . . . × Up × Ωn ⊆ U n=p+1 10. Show the existence of some Vx ∈ H such that x ∈ Vx ⊆ U . 11. Show that H is a countable base of the topological space (Ω, T ). ∞ 12. Show that ⊗+=1 B (Ωn ) ⊆ B (Ω). n ∞ 13. Show that H ⊆ ⊗+=1 B (Ωn ). n ∞ 14. Show that B (Ω) = ⊗+=1 B (Ωn ) n Tutorial 6: Product Spaces 22 Theorem 27 Let (Ωn , Tn )n≥1 be a sequence of topological spaces ∞ ∞ with countable base. Then, the product space (Π+=1 Ωn , +=1 Tn ) has n n a countable base and: +∞ B +∞ Ωn n=1 B (Ωn ) = n=1 Exercise 18. 1. Show that if (Ω, T ) has a countable base and n ≥ 1: B (Ωn ) = B (Ω) ⊗ . . . ⊗ B (Ω) n ¯n ¯ ¯ 2. Show that B (R ) = B (R) ⊗ . . . ⊗ B (R). 3. Show that B (C) = B (R) ⊗ B (R). Tutorial 6: Product Spaces 23 Definition 58 We say that a metric space (E, d) is separable, if and only if there exists a finite or countable dense subset of E , i.e. ¯ ¯ a finite or countable subset A of E such that E = A, where A is the closure of A in E . Exercise 19. Let (E, d) be a metric space. 1. Suppose that (E, d) is separable. Let H = {B (xn , 1 ) : n, p ≥ 1}, p where {xn : n ≥ 1} is a countable dense subset in E . Show that d H is a countable base of the metric topological space (E, TE ). d 2. Suppose conversely that (E, TE ) has a countable base H. For all V ∈ H such that V = ∅, take xV ∈ V . Show that the set {xV : V ∈ H , V = ∅} is at most countable and dense in E . 3. For all x, y, x , y ∈ E , show that: |d(x, y ) − d(x , y )| ≤ d(x, x ) + d(y, y ) Tutorial 6: Product Spaces 24 4. Let TE ×E be the product topology on E × E . Show that the map d : (E × E, TE ×E ) → (R+ , TR+ ) is continuous. ¯ ¯ 5. Show that d : (E × E, B (E × E )) → (R, B (R)) is measurable. ¯ ¯ 6. Show that d : (E × E, B (E ) ⊗B (E )) → (R, B (R)) is measurable, whenever (E, d) is a separable metric space. 7. Let (Ω, F ) be a measurable space and f, g : (Ω, F ) → (E, B (E )) be measurable maps. Show that Φ : (Ω, F ) → E × E defined by Φ(ω) = (f (ω ), g (ω)) is measurable with respect to the product σ -algebra B (E ) ⊗ B (E ). ¯ ¯ 8. Show that if (E, d) is separable, then Ψ : (Ω, F ) → (R, B (R)) defined by Ψ(ω) = d(f (ω ), g (ω )) is measurable. 9. Show that if (E, d) is separable then {f = g } ∈ F . 10. Let (En , dn )n≥1 be a sequence of separable metric spaces. Show ∞ that the product space Π+=1 En is metrizable and separable. n Tutorial 6: Product Spaces 25 Exercise 20. Prove the following theorem. Theorem 28 Let (Ωi , Fi )i∈I be a family of measurable spaces and (Ω, F ) be a measurable space. For all i ∈ I , let fi : Ω → Ωi be a map, and define f : Ω → Πi∈I Ωi by f (ω ) = (fi (ω ))i∈I . Then, the map: f : (Ω, F ) → Fi Ωi , i∈I i∈I is measurable, if and only if each fi : (Ω, F ) → (Ωi , Fi ) is measurable. Exercise 21. 1. Let φ, ψ : R2 → R with φ(x, y ) = x + y and ψ (x, y ) = x.y . Show that both φ and ψ are continuous. ¯ ¯ 2. Show that φ, ψ : (R2 , B (R) ⊗B (R))→(R, B (R)) are measurable. 3. Let (Ω, F ) be a measurable space, and f, g : (Ω, F ) → (R, B (R)) be measurable maps. Using the previous results, show that f + g and f.g are measurable with respect to F and B (R). Tutorial 7: Fubini Theorem 1 7. Fubini Theorem Definition 59 Let (Ω1 , F1 ) and (Ω2 , F2 ) be two measurable spaces. Let E ⊆ Ω1 × Ω2 . For all ω1 ∈ Ω1 , we call ω1 -section of E in Ω2 , the set: E ω1 = {ω2 ∈ Ω2 : (ω 1 , ω 2 ) ∈ E } Exercise 1. Let (Ω1 , F1 ) and (Ω2 , F2 ) be two measurable spaces. Given ω 1 ∈ Ω1 , define: Γ ω 1 = { E ⊆ Ω1 × Ω 2 , E ω 1 ∈ F 2 } 1. Show that for all ω 1 ∈ Ω1 , Γω1 is a σ -algebra on Ω1 × Ω2 . 2. Show that for all ω 1 ∈ Ω1 , F1 F 2 ⊆ Γω 1 . 3. Show that for all ω 1 ∈ Ω1 and E ∈ F1 ⊗ F2 , we have E ω1 ∈ F2 . 4. Show that the map ω → 1E (ω1 , ω) is measurable with respect ¯ to F2 and B (R), for all ω 1 ∈ Ω1 and E ∈ F1 ⊗ F2 . Tutorial 7: Fubini Theorem 2 5. Let s be a simple function on (Ω1 × Ω2 , F1 ⊗ F2 ). Show that for all ω 1 ∈ Ω1 , the map ω → s(ω 1 , ω ) is measurable with respect ¯ to F2 and B (R). 6. Let f : (Ω1 × Ω2 , F1 ⊗ F2 ) → [0, +∞] be a non-negative, measurable map. Show that for all ω 1 ∈ Ω1 , the map ω → f (ω 1 , ω) ¯ is measurable with respect to F2 and B (R). ¯ ¯ 7. Let f : (Ω1 × Ω2 , F1 ⊗ F2 ) → (R, B (R)) be a measurable map. Show that for all ω 1 ∈ Ω1 , the map ω → f (ω 1 , ω) is measurable ¯ with respect to F2 and B (R). 8. Show the following theorem: Theorem 29 Let (E, d) be a metric space, and (Ω1 , F1 ), (Ω2 , F2 ) be two measurable spaces. Let f : (Ω1 × Ω2 , F1 ⊗ F2 ) → (E, B (E )) be a measurable map . Then for all ω 1 ∈ Ω1 , the map ω → f (ω 1 , ω ) is measurable with respect to F2 and B (E ). Tutorial 7: Fubini Theorem 3 Exercise 2. Let (Ωi , Fi )i∈I be a family of measurable spaces with cardI ≥ 2. Let f : (Πi∈I Ωi , ⊗i∈I Fi ) → (E, B (E )) be a measurable map, where (E, d) be a metric space. Let i1 ∈ I . Put E1 = Ωi1 , E1 = Fi1 , E2 = Πi∈I \{i1 } Ωi , E2 = ⊗i∈I \{i1 } Fi . 1. Explain why f can be viewed as a map defined on E1 × E2 . 2. Show that f : (E1 × E2 , E1 ⊗ E2 ) → (E, B (E )) is measurable. 3. For all ωi1 ∈ Ωi1 , show that the map ω → f (ω i1 , ω) defined on Πi∈I \{i1 } Ωi is measurable w.r. to ⊗i∈I \{i1 } Fi and B (E ). Definition 60 Let (Ω, F , µ) be a measure space. (Ω, F , µ) is said to be a finite measure space, or we say that µ is a finite measure, if and only if µ(Ω) < +∞. Tutorial 7: Fubini Theorem 4 Definition 61 Let (Ω, F , µ) be a measure space. (Ω, F , µ) is said to be a σ -finite measure space, or µ a σ -finite measure, if and only if there exists a sequence (Ωn )n≥1 in F such that Ωn ↑ Ω and µ(Ωn ) < +∞, for all n ≥ 1. Exercise 3. Let (Ω, F , µ) be a measure space. 1. Show that (Ω, F , µ) is σ -finite if and only if there exists a se∞ quence (Ωn )n≥1 in F such that Ω = +=1 Ωn , and µ(Ωn ) < +∞ n for all n ≥ 1. 2. Show that if (Ω, F , µ) is finite, then µ has values in R+ . 3. Show that if (Ω, F , µ) is finite, then it is σ -finite. 4. Let F : R → R be a right-continuous, non-decreasing map. Show that the measure space (R, B (R), dF ) is σ -finite, where dF is the stieltjes measure associated with F . Tutorial 7: Fubini Theorem 5 Exercise 4. Let (Ω1 , F1 ) be a measurable space, and (Ω2 , F2 , µ2 ) be a σ -finite measure space. For all E ∈ F1 ⊗ F2 and ω 1 ∈ Ω1 , define: ΦE (ω 1 ) = 1E (ω 1 , x)dµ2 (x) Ω2 Let D be the set of subsets of Ω1 × Ω2 , defined by: ¯ ¯ D = {E ∈ F1 ⊗ F2 : ΦE : (Ω1 , F1 ) → (R, B (R)) is measurable} 1. Explain why for all E ∈ F1 ⊗ F2 , the map ΦE is well defined. 2. Show that F1 F2 ⊆ D . 3. Show that if µ2 is finite, A, B ∈ D and A ⊆ B , then B \ A ∈ D. 4. Show that if En ∈ F1 ⊗ F2 , n ≥ 1 and En ↑ E , then ΦEn ↑ ΦE . 5. Show that if µ2 is finite then D is a dynkin system on Ω1 × Ω2 . ¯ ¯ 6. Show that if µ2 is finite, then the map ΦE : (Ω1 , F1 ) → (R, B (R)) is measurable, for all E ∈ F1 ⊗ F2 . Tutorial 7: Fubini Theorem 6 7. Let (Ωn )n≥1 in F2 be such that Ωn ↑ Ω2 and µ2 (Ωn ) < +∞. 2 2 2 Ωn Define µn = µ2 2 = µ2 (• ∩ Ωn ). For E ∈ F1 ⊗ F2 , we put: 2 2 Φn (ω 1 ) = E 1E (ω 1 , x)dµn (x) 2 Ω2 Show that Φn E ¯ ¯ : (Ω1 , F1 ) → (R, B (R)) is measurable, and: Φn (ω 1 ) = E Ω2 1Ωn (x)1E (ω 1 , x)dµ2 (x) 2 Deduce that Φn ↑ ΦE . E ¯ ¯ 8. Show that the map ΦE : (Ω1 , F1 ) → (R, B (R)) is measurable, for all E ∈ F1 ⊗ F2 . 9. Let s be a simple function on (Ω1 × Ω2 , F1 ⊗ F2 ). Show that the map ω → Ω2 s(ω, x)dµ2 (x) is well defined and measurable ¯ with respect to F1 and B (R). 10. Show the following theorem: Tutorial 7: Fubini Theorem 7 Theorem 30 Let (Ω1 , F1 ) be a measurable space, and (Ω2 , F2 , µ2 ) be a σ -finite measure space. Then for all non-negative and measurable map f : (Ω1 × Ω2 , F1 ⊗ F2 ) → [0, +∞], the map: ω→ f (ω, x)dµ2 (x) Ω2 ¯ is measurable with respect to F1 and B (R). Exercise 5. Let (Ωi , Fi )i∈I be a family of measurable spaces, with cardI ≥ 2. Let i0 ∈ I , and suppose that µ0 is a σ -finite measure on (Ωi0 , Fi0 ). Show that if f : (Πi∈I Ωi , ⊗i∈I Fi ) → [0, +∞] is a nonnegative and measurable map, then: ω→ f (ω, x)dµ0 (x) Ωi0 ¯ defined on Πi∈I \{i0 } Ωi , is measurable w.r. to ⊗i∈I \{i0 } Fi and B (R). Tutorial 7: Fubini Theorem 8 Exercise 6. Let (Ω1 , F1 , µ1 ) and (Ω2 , F2 , µ2 ) be two σ -finite measure spaces. For all E ∈ F1 ⊗ F2 , we define: µ1 ⊗ µ2 (E ) = 1E (x, y )dµ2 (y ) dµ1 (x) Ω1 Ω2 1. Explain why µ1 ⊗ µ2 : F1 ⊗ F2 → [0, +∞] is well defined. 2. Show that µ1 ⊗ µ2 is a measure on F1 ⊗ F2 . 3. Show that if A × B ∈ F1 F2 , then: µ1 ⊗ µ2 (A × B ) = µ1 (A)µ2 (B ) Exercise 7. Further to ex. (6), suppose that µ : F1 ⊗ F2 → [0, +∞] is another measure on F1 ⊗ F2 with µ(A × B ) = µ1 (A)µ2 (B ), for all measurable rectangle A × B . Let (Ωn )n≥1 and (Ωn )n≥1 be sequences 1 2 in F1 and F2 respectively, such that Ωn ↑ Ω1 , Ωn ↑ Ω2 , µ1 (Ωn ) < +∞ 1 2 1 and µ2 (Ωn ) < +∞. Define, for all n ≥ 1: 2 Dn = {E ∈ F1 ⊗ F2 : µ(E ∩ (Ωn × Ωn )) = µ1 ⊗ µ2 (E ∩ (Ωn × Ωn ))} 1 2 1 2 Tutorial 7: Fubini Theorem 9 1. Show that for all n ≥ 1, F1 F2 ⊆ Dn . 2. Show that for all n ≥ 1, Dn is a dynkin system on Ω1 × Ω2 . 3. Show that µ = µ1 ⊗ µ2 . 4. Show that (Ω1 × Ω2 , F1 ⊗F2 , µ1 ⊗ µ2 ) is a σ -finite measure space. 5. Show that for all E ∈ F1 ⊗ F2 , we have: µ1 ⊗ µ2 (E ) = 1E (x, y )dµ1 (x) dµ2 (y ) Ω2 Ω1 Exercise 8. Let (Ω1 , F1 , µ1 ), . . . , (Ωn , Fn , µn ) be n σ -finite measure spaces, n ≥ 2. Let i0 ∈ {1, . . . , n} and put E1 = Ωi0 , E2 = Πi=i0 Ωi , E1 = Fi0 and E2 = ⊗i=i0 Fi . Put ν1 = µi0 , and suppose that ν2 is a σ -finite measure on (E2 , E2 ) such that for all measurable rectangle Πi=i0 Ai ∈ i=i0 Fi , we have ν2 (Πi=i0 Ai ) = Πi=i0 µi (Ai ). Tutorial 7: Fubini Theorem 10 1. Show that ν1 ⊗ ν2 is a σ -finite measure on the measure space (Ω1 × . . . × Ωn , F1 ⊗ . . . ⊗ Fn ) such that for all measurable rectangles A1 × . . . × An , we have: ν1 ⊗ ν2 (A1 × . . . × An ) = µ1 (A1 ) . . . µn (An ) 2. Show by induction the existence of a measure µ on F1 ⊗ . . . ⊗Fn , such that for all measurable rectangles A1 × . . . × An , we have: µ(A1 × . . . × An ) = µ1 (A1 ) . . . µn (An ) 3. Show the uniqueness of such measure, denoted µ1 ⊗ . . . ⊗ µn . 4. Show that µ1 ⊗ . . . ⊗ µn is σ -finite. 5. Let i0 ∈ {1, . . . , n}. Show that µi0 ⊗ (⊗i=i0 µi ) = µ1 ⊗ . . . ⊗ µn . Tutorial 7: Fubini Theorem 11 Definition 62 Let (Ω1 , F1 , µ1 ), . . . , (Ωn , Fn , µn ) be n σ -finite measure spaces, with n ≥ 2. We call product measure of µ1 , . . . , µn , the unique measure on F1 ⊗ . . . ⊗ Fn , denoted µ1 ⊗ . . . ⊗ µn , such that for all measurable rectangles A1 × . . . × An in F1 . . . Fn , we have: µ1 ⊗ . . . ⊗ µn (A1 × . . . × An ) = µ1 (A1 ) . . . µn (An ) This measure is itself σ -finite. Exercise 9. Prove that the following definition is legitimate: Definition 63 We call lebesgue measure in Rn , n ≥ 1, the unique measure on (Rn , B (Rn )), denoted dx, dxn or dx1 . . . dxn , such that for all ai ≤ bi , i = 1, . . . , n, we have: n dx([a1 , b1 ] × . . . × [an , bn ]) = (bi − ai ) i=1 Tutorial 7: Fubini Theorem 12 Exercise 10. 1. Show that (Rn , B (Rn ), dxn ) is a σ -finite measure space. 2. For n, p ≥ 1, show that dxn+p = dxn ⊗ dxp . Exercise 11. Let (Ω1 , F1 , µ1 ) and (Ω2 , F2 , µ2 ) be σ -finite. 1. Let s be a simple function on (Ω1 × Ω2 , F1 ⊗ F2 ). Show that: sdµ1 ⊗ µ2 = Ω1 ×Ω2 sdµ2 dµ1 = Ω1 Ω2 sdµ1 dµ2 Ω2 Ω1 2. Show the following: Theorem 31 (Fubini) Let (Ω1 , F1 , µ1 ) and (Ω2 , F2 , µ2 ) be two σ finite measure spaces. Let f : (Ω1 × Ω2 , F1 ⊗ F2 ) → [0, +∞] be a non-negative and measurable map. Then: f dµ1 ⊗ µ2 = Ω1 ×Ω2 f dµ2 dµ1 = Ω1 Ω2 f dµ1 dµ2 Ω2 Ω1 Tutorial 7: Fubini Theorem 13 Exercise 12. Let (Ω1 , F1 , µ1 ), . . . , (Ωn , Fn , µn ) be n σ -finite measure spaces, n ≥ 2. Let f : (Ω1 × . . . × Ωn , F1 ⊗ . . . ⊗ Fn ) → [0, +∞] be a non-negative, measurable map. Let σ be a permutation of Nn , i.e. a bijection from Nn to itself. 1. For all ω ∈ Πi=σ(1) Ωi , define: J1 (ω ) = f (ω, x)dµσ(1) (x) Ωσ(1) Explain why J1 : (Πi=σ(1) Ωi , ⊗i=σ(1) Fi ) → [0, +∞] is a well defined, non-negative and measurable map. 2. Suppose Jk : (Πi∈{σ(1),...,σ(k)} Ωi , ⊗i∈{σ(1),...,σ(k)} Fi ) → [0, +∞] is a non-negative, measurable map, for 1 ≤ k < n − 2. Define: Jk+1 (ω ) = Jk (ω, x)dµσ(k+1) (x) Ωσ(k+1) and show that: Jk+1 : (Πi∈{σ(1),...,σ(k+1)} Ωi , ⊗i∈{σ(1),...,σ(k+1)} Fi ) → [0, +∞] Tutorial 7: Fubini Theorem 14 is also well-defined, non-negative and measurable. 3. Propose a rigorous definition for the following notation: ... Ωσ(n) f dµσ(1) . . . dµσ(n) Ωσ(1) Exercise 13. Further to ex. (12), Let (fp )p≥1 be a sequence of nonnegative and measurable maps: fp : (Ω1 × . . . × Ωn , F1 ⊗ . . . ⊗ Fn ) → [0, +∞] such that fp ↑ f . Define similarly: p J1 (ω ) = fp (ω, x)dµσ(1) (x) Ωσ(1) p Jk+1 (ω ) 1. Show that p Jk (ω, x)dµσ(k+1) (x) , 1 ≤ k < n − 2 = Ωσ(k+1) p J1 ↑ J1 . Tutorial 7: Fubini Theorem 15 p p 2. Show that if Jk ↑ Jk , then Jk+1 ↑ Jk+1 , 1 ≤ k < n − 2. 3. Show that: ... Ωσ(n) fp dµσ(1) Ωσ(1) . . . dµσ(n) ↑ ... Ωσ(n) f dµσ(1) . . . dµσ(n) Ωσ(1) 4. Show that the map µ : F1 ⊗ . . . ⊗ Fn → [0, +∞], defined by: ... µ(E ) = Ωσ(n) 1E dµσ(1) . . . dµσ(n) Ωσ(1) is a measure on F1 ⊗ . . . ⊗ Fn . 5. Show that for all E ∈ F1 ⊗ . . . ⊗ Fn , we have: µ1 ⊗ . . . ⊗ µn (E ) = ... Ωσ(n) 6. Show the following: 1E dµσ(1) . . . dµσ(n) Ωσ(1) Tutorial 7: Fubini Theorem 16 Theorem 32 Let (Ω1 , F1 , µ1 ), . . . , (Ωn , Fn , µn ) be n σ -finite measure spaces, with n ≥ 2. Let f : (Ω1 ×. . .×Ωn , F1 ⊗. . .⊗Fn ) → [0, +∞] be a non-negative and measurable map. let σ be a permutation of Nn . Then: f dµ1 ⊗ . . . ⊗ µn = Ω1 ×...×Ωn ... Ωσ(n) f dµσ(1) . . . dµσ(n) Ωσ(1) Exercise 14. Let (Ω, F , µ) be a measure space. Define: ¯ L1 = {f : Ω → R , ∃g ∈ L1 (Ω, F , µ) , f = g µ-a.s.} R 1. Show that if f ∈ L1 , then |f | < +∞, µ-a.s. 2. Suppose there exists A ⊆ Ω, such that A ∈ F and A ⊆ N for some N ∈ F with µ(N ) = 0. Show that 1A ∈ L1 and 1A is not ¯ measurable with respect to F and B (R). 3. Explain why if f ∈ L1 , the integrals be well defined. |f |dµ and f dµ may not Tutorial 7: Fubini Theorem 17 ¯ ¯ 4. Suppose that f : (Ω, F ) → (R, B (R)) is a measurable map with 1 |f |dµ < +∞. Show that f ∈ L . 5. Show that if f ∈ L1 and f = f1 µ-a.s. then f1 ∈ L1 . 6. Suppose that f ∈ L1 and g1 , g2 ∈ L1 (Ω, F , µ) are such that R f = g1 µ-a.s. and f = g2 µ-a.s.. Show that g1 dµ = g2 dµ. 7. Propose a definition of the integral f dµ for f ∈ L1 which extends the integral defined on L1 (Ω, F , µ). R Exercise 15. Further to ex. (14), Let (fn )n≥1 be a sequence in L1 , and f, h ∈ L1 , with fn → f µ-a.s. and for all n ≥ 1, |fn | ≤ h µ-a.s.. 1. Show the existence of N1 ∈ F , µ(N1 ) = 0, such that for all c ω ∈ N1 , fn (ω ) → f (ω ), and for all n ≥ 1, |fn (ω )| ≤ h(ω ). 2. Show the existence of gn , g, h1 ∈ L1 (Ω, F , µ) and N2 ∈ F , R c µ(N2 ) = 0, such that for all ω ∈ N2 , g (ω ) = f (ω ), h(ω ) = h1 (ω ), and for all n ≥ 1, gn (ω ) = fn (ω ). Tutorial 7: Fubini Theorem 18 3. Show the existence of N ∈ F , µ(N ) = 0, such that for all ω ∈ N c , gn (ω ) → g (ω ), and for all n ≥ 1, |gn (ω )| ≤ h1 (ω ). 4. Show that the Dominated Convergence Theorem can be applied to gn 1N c , g 1N c and h1 1N c . 5. Recall the definition of 6. Show that |fn − f |dµ when f, fn ∈ L1 . |fn − f |dµ → 0. Exercise 16. Let (Ω1 , F1 , µ1 ) and (Ω2 , F2 , µ2 ) be two σ -finite measure spaces. let f be an element of L1 (Ω1 × Ω2 , F1 ⊗ F2 , µ1 ⊗ µ2 ). R 1. Let A = {ω 1 ∈ Ω1 : Ω2 |f (ω 1 , x)|dµ2 (x) < +∞}. Show that A ∈ F1 and µ1 (Ac ) = 0. 2. Show that f (ω1 , .) ∈ L1 (Ω2 , F2 , µ2 ) for all ω 1 ∈ A. R ¯ 3. Show that I (ω 1 ) = Ω2 f (ω 1 , x)dµ2 (x) is well defined for all ¯ ω1 ∈ A. Let I be an arbitrary extension of I , on Ω1 . Tutorial 7: Fubini Theorem 19 4. Define J = I 1A . Show that: f − (ω, x)dµ2 (x) f + (ω, x)dµ2 (x) − 1A (ω ) J (ω ) = 1A (ω ) Ω2 Ω2 5. Show that J is F1 -measurable and R-valued. 6. Show that J ∈ L1 (Ω1 , F1 , µ1 ) and that J = I µ1 -a.s. R 7. Propose a definition for the integral: f (x, y )dµ2 (y ) dµ1 (x) Ω1 8. Show that Ω1 (1A Ω2 Ω2 f + dµ2 )dµ1 = Ω1 ×Ω2 f + dµ1 ⊗ µ2 . 9. Show that: f (x, y )dµ2 (y ) dµ1 (x) = Ω1 Ω2 10. Prove the following: f dµ1 ⊗ µ2 Ω1 ×Ω2 Tutorial 7: Fubini Theorem 20 Theorem 33 Let (Ω1 , F1 , µ1 ) and (Ω2 , F2 , µ2 ) be two σ -finite measure spaces. Let f ∈ L1 (Ω1 × Ω2 , F1 ⊗ F2 , µ1 ⊗ µ2 ). Then, the map: C ω1 → f (ω1 , x)dµ2 (x) Ω2 is µ1 -almost surely equal to an element of L1 (Ω1 , F1 , µ1 ) and: C f (x, y )dµ2 (y ) dµ1 (x) = Ω1 Ω2 f dµ1 ⊗ µ2 Ω1 ×Ω2 Exercise 17. Let (Ω1 , F1 , µ1 ),. . . ,(Ωn , Fn , µn ) be n σ -finite measure spaces, n ≥ 2. Let f ∈ L1 (Ω1 × . . . × Ωn , F1 ⊗ . . . ⊗ Fn , µ1 ⊗ . . . ⊗ µn ). C Let σ be a permutation of Nn . 1. For all ω ∈ Πi=σ(1) Ωi , define: J1 (ω ) = f (ω, x)dµσ(1) (x) Ωσ(1) Tutorial 7: Fubini Theorem 21 Explain why J1 is well defined and equal to an element of L1 (Πi=σ(1) Ωi , ⊗i=σ(1) Fi , ⊗i=σ(1) µi ), ⊗i=σ(1) µi -almost surely. C ¯ 2. Suppose 1 ≤ k < n − 2 and that Jk is well defined and equal to an element of: L1 (Πi∈{σ(1),...,σ(k)} Ωi , ⊗i∈{σ(1),...,σ(k)} Fi , ⊗i∈{σ(1),...,σ(k)} µi ) C ⊗i∈{σ(1),...,σ(k)} µi -almost surely. Define: ¯ Jk (ω, x)dµσ(k+1) (x) Jk+1 (ω) = Ωσ(k+1) What can you say about Jk+1 . 3. Show that: ... Ωσ(n) f dµσ(1) . . . dµσ(n) Ωσ(1) is a well defined complex number. (Propose a definition for it). Tutorial 7: Fubini Theorem 22 4. Show that: ... Ωσ(n) f dµσ(1) . . . dµσ(n) = Ωσ(1) f dµ1 ⊗ . . . ⊗ µn Ω1 ×...×Ωn Tutorial 8: Jensen inequality 1 8. Jensen inequality ¯ Definition 64 Let a, b ∈ R, with a < b. Let φ : ]a, b[→ R be an R-valued function. We say that φ is a convex function, if and only if, for all x, y ∈]a, b[ and t ∈ [0, 1], we have: φ(tx + (1 − t)y ) ≤ tφ(x) + (1 − t)φ(y ) ¯ Exercise 1. Let a, b ∈ R, with a < b. Let φ : ]a, b[→ R be a map. 1. Show that φ : ]a, b[→ R is convex, if and only if for all x1 , . . . , xn in ]a, b[ and α1 , . . . , αn in R+ with α1 + . . . + αn = 1, n ≥ 1, we have: φ(α1 x1 + . . . + αn xn ) ≤ α1 φ(x1 ) + . . . αn φ(xn ) 2. Show that φ : ]a, b[→ R is convex, if and only if for all x, y, z with a < x < y < z < b we have: y−x z−y φ(x) + φ(z ) φ(y ) ≤ z−x z−x Tutorial 8: Jensen inequality 2 3. Show that φ : ]a, b[→ R is convex if and only if for all x, y, z with a < x < y < z < b, we have: φ(z ) − φ(y ) φ(y ) − φ(x) ≤ y−x z−y 4. Let φ : ]a, b[→ R be convex. Let x0 ∈]a, b[, and u, u , v, v ∈]a, b[ be such that u < u < x0 < v < v . Show that for all x ∈]x0 , v [: φ(x) − φ(x0 ) φ(v ) − φ(v ) φ(u ) − φ(u) ≤ ≤ u −u x − x0 v −v and deduce that limx↓↓x0 φ(x) = φ(x0 ) 5. Show that if φ : ]a, b[→ R is convex, then φ is continuous. 6. Define φ : [0, 1] → R by φ(0) = 1 and φ(x) = 0 for all x ∈]0, 1]. Show that φ(tx + (1 − t)y ) ≤ tφ(x) + (1 − t)φ(y ), ∀x, y, t ∈ [0, 1], but that φ fails to be continuous on [0, 1]. Tutorial 8: Jensen inequality 3 Definition 65 Let (Ω, T ) be a topological space. We say that (Ω, T ) is a compact topological space if and only if, for all family (Vi )i∈I of open sets in Ω, such that Ω = ∪i∈I Vi , there exists a finite subset {i1 , . . . , in } of I such that Ω = Vi1 ∪ . . . ∪ Vin . In short, we say that (Ω, T ) is compact if and only if, from any open covering of Ω, one can extract a finite sub-covering. Definition 66 Let (Ω, T ) be a topological space, and K ⊆ Ω. We say that K is a compact subset of Ω, if and only if the induced topological space (K, T|K ) is a compact topological space. Exercise 2. Let (Ω, T ) be a topological space. 1. Show that if (Ω, T ) is compact, it is a compact subset of itself. 2. Show that ∅ is a compact subset of Ω. 3. Show that if Ω ⊆ Ω and K is a compact subset of Ω , then K is also a compact subset of Ω. Tutorial 8: Jensen inequality 4 4. Show that if (Vi )i∈I is a family of open sets in Ω such that K ⊆ ∪i∈I Vi , then K = ∪i∈I (Vi ∩ K ) and Vi ∩ K is open in K for all i ∈ I . 5. Show that K ⊆ Ω is a compact subset of Ω, if and only if for any family (Vi )i∈I of open sets in Ω such that K ⊆ ∪i∈I Vi , there is a finite subset {i1 , . . . , in } of I such that K ⊆ Vi1 ∪ . . . ∪ Vin . 6. Show that if (Ω, T ) is compact and K is closed in Ω, then K is a compact subset of Ω. Exercise 3. Let a, b ∈ R, a < b. Let (Vi )i∈I be a family of open sets in R such that [a, b] ⊆ ∪i∈I Vi . We define A as the set of all x ∈ [a, b] such that [a, x] can be covered by a finite number of Vi ’s. Let c = sup A. 1. Show that a ∈ A. 2. Show that there is > 0 such that a + ∈ A. Tutorial 8: Jensen inequality 5 3. Show that a < c ≤ b. 4. Show the existence of i0 ∈ I and c , c with a < c < c < c , such that ]c , c ] ⊆ Vi0 . 5. Show that [a, c ] can be covered by a finite number of Vi ’s. 6. Show that [a, c ] can be covered by a finite number of Vi ’s. 7. Show that b ∧ c ≤ c and conclude that c = b. 8. Show that [a, b] is a compact subset of R. Theorem 34 Let a, b ∈ R, a < b. The closed interval [a, b] is a compact subset of R. Definition 67 Let (Ω, T ) be a topological space. We say that (Ω, T ) is a hausdorff topological space, if and only if for all x, y ∈ Ω with x = y , there exists open sets U and V in Ω, such that: x∈U , y ∈V , U ∩V =∅ Tutorial 8: Jensen inequality 6 Exercise 4. Let (Ω, T ) be a topological space. 1. Show that if (Ω, T ) is hausdorff and Ω ⊆ Ω, then the induced topological space (Ω , T|Ω ) is itself hausdorff. 2. Show that if (Ω, T ) is metrizable, then it is hausdorff. ¯ 3. Show that any subset of R is hausdorff. 4. Let (Ωi , Ti )i∈I be a family of hausdorff topological spaces. Show that the product topological space Πi∈I Ωi is hausdorff. Exercise 5. Let (Ω, T ) be a hausdorff topological space. Let K be a compact subset of Ω and suppose there exists y ∈ K c . 1. Show that for all x ∈ K , there are open sets Vx , Wx in Ω, such that y ∈ Vx , x ∈ Wx and Vx ∩ Wx = ∅. 2. Show that there exists a finite subset {x1 , . . . , xn } of K such that K ⊆ W y where W y = Wx1 ∪ . . . ∪ Wxn . Tutorial 8: Jensen inequality 7 3. Let V y = Vx1 ∩ . . . ∩ Vxn . Show that V y is open and V y ∩ W y = ∅. 4. Show that y ∈ V y ⊆ K c . 5. Show that K c = ∪y∈K c V y 6. Show that K is closed in Ω. Theorem 35 Let (Ω, T ) be a hausdorff topological space. For all K ⊆ Ω, if K is a compact subset, then it is closed. Definition 68 Let (E, d) be a metric space. For all A ⊆ E , we ¯ call diameter of A with respect to d, the element of R denoted δ (A), defined as δ (A) = sup{d(x, y ) : x, y ∈ A}, with the convention that δ (∅) = −∞. Definition 69 Let (E, d) be a metric space, and A ⊆ E . We say that A is bounded, if and only if its diameter is finite, i.e. δ (A) < +∞. Tutorial 8: Jensen inequality 8 Exercise 6. Let (E, d) be a metric space. Let A ⊆ E . 1. Show that δ (A) = 0 if and only if A = {x} for some x ∈ E . ¯ 2. Let φ : R → [−1, 1] be an increasing homeomorphism. Define d (x, y ) = |x − y | and d (x, y ) = |φ(x) − φ(y )|, for all x, y ∈ R. Show that d is a metric on R inducing the usual topology on R. Show that R is bounded with respect to d but not with respect to d . 3. Show that if K ⊆ E is a compact subset of E , for all there is a finite subset {x1 , . . . , xn } of K such that: > 0, K ⊆ B (x1 , ) ∪ . . . ∪ B (xn , ) 4. Show that any compact subset of any metrizable topological space (Ω, T ), is bounded with respect to any metric inducing the topology T . Tutorial 8: Jensen inequality 9 Exercise 7. Suppose K is a closed subset of R which is bounded with respect to the usual metric on R. 1. Show that there exists M ∈ R+ such that K ⊆ [−M, M ]. 2. Show that K is also closed in [−M, M ]. 3. Show that K is a compact subset of [−M, M ]. 4. Show that K is a compact subset of R. 5. Show that any compact subset of R is closed and bounded. 6. Show the following: Theorem 36 A subset of R is compact if and only if it is closed, and bounded with respect to the usual metric on R. Tutorial 8: Jensen inequality 10 Exercise 8. Let (Ω, T ) and (S, TS ) be two topological spaces. Let f : (Ω, T ) → (S, TS ) be a continuous map. 1. Show that if (Wi )i∈I is an open covering of f (Ω), then the family (f −1 (Wi ))i∈I is an open covering of Ω. 2. Show that if (Ω, T ) is a compact topological space, then f (Ω) is a compact subset of (S, TS ). Exercise 9. ¯¯ 1. Show that (R, TR ) is a compact topological space. ¯ 2. Show that any compact subset of R is a compact subset of R. ¯ 3. Show that a subset of R is compact if and only if it is closed. ¯ 4. Let A be a non-empty subset of R, and let α = sup A. Show that if α = −∞, then for all U ∈ TR with α ∈ U , there exists ¯ ¯ β ∈ R with β < α and ]β, α] ⊆ U . Conclude that α ∈ A. Tutorial 8: Jensen inequality 11 ¯ 5. Show that if A is a non-empty closed subset of R, then we have sup A ∈ A and inf A ∈ A. 6. Consider A = {x ∈ R , sin(x) = 0}. Show that A is closed in R, but that sup A ∈ A and inf A ∈ A. 7. Show that if A is a non-empty, closed and bounded subset of R, then sup A ∈ A and inf A ∈ A. Exercise 10. Let (Ω, T ) be a compact, non-empty topological space. ¯¯ Let f : (Ω, T ) → (R, TR ) be a continuous map. 1. Show that if f (Ω) ⊆ R, the continuity of f with respect to TR ¯ is equivalent to the continuity of f with respect to TR . 2. Show the following: Tutorial 8: Jensen inequality 12 ¯¯ Theorem 37 Let f : (Ω, T ) → (R, TR ) be a continuous map, where (Ω, T ) is a non-empty topological space. Then, if (Ω, T ) is compact, f attains its maximum and minimum, i.e. there exist xm , xM ∈ Ω, such that: f (xm ) = inf f (x) , f (xM ) = sup f (x) x ∈Ω x ∈Ω Exercise 11. Let a, b ∈ R, a < b. Let f : [a, b] → R be continuous on [a, b], and differentiable on ]a, b[, with f (a) = f (b). 1. Show that if c ∈]a, b[ and f (c) = supx∈[a,b] f (x), then f (c) = 0. 2. Show the following: Theorem 38 (Rolle) Let a, b ∈ R, a < b. Let f : [a, b] → R be continuous on [a, b], and differentiable on ]a, b[, with f (a) = f (b). Then, there exists c ∈]a, b[ such that f (c) = 0. Tutorial 8: Jensen inequality 13 Exercise 12. Let a, b ∈ R, a < b. Let f : [a, b] → R be continuous on [a, b] and differentiable on ]a, b[. Define: h(x) = f (x) − (x − a) f (b) − f (a) b−a 1. Show that h is continuous on [a, b] and differentiable on ]a, b[. 2. Show the existence of c ∈]a, b[ such that: f (b) − f (a) = (b − a)f (c) Exercise 13. Let a, b ∈ R, a < b. Let f : [a, b] → R be a map. Let n ≥ 0. We assume that f is of class C n on [a, b], and that f (n+1) exists on ]a, b[. Define: n h(x) = f (b) − f (x) − k=1 (b − x)n+1 (b − x)k (k) f (x) − α k! (n + 1)! where α is chosen such that h(a) = 0. Tutorial 8: Jensen inequality 14 1. Show that h is continuous on [a, b] and differentiable on ]a, b[. 2. Show that for all x ∈]a, b[: h (x) = (b − x)n (α − f (n+1) (x)) n! 3. Prove the following: Theorem 39 (Taylor-Lagrange) Let a, b ∈ R, a < b, and n ≥ 0. Let f : [a, b] → R be a map of class C n on [a, b] such that f (n+1) exists on ]a, b[. Then, there exists c ∈]a, b[ such that: n f (b) − f (a) = k=1 (b − a)n+1 (n+1) (b − a)k (k) f (a) + f (c) k! (n + 1)! Tutorial 8: Jensen inequality 15 ¯ Exercise 14. Let a, b ∈ R, a < b and φ : ]a, b[→ R be differentiable. 1. Show that if φ is convex, then for all x, y ∈]a, b[, x < y , we have: φ (x) ≤ φ (y ) 2. Show that if x, y, z ∈]a, b[ with x < y < z , there are c1 , c2 ∈]a, b[, with c1 < c2 and: φ(y ) − φ(x) φ(z ) − φ(y ) = φ (c1 )(y − x) = φ (c2 )(z − y ) 3. Show conversely that if φ is non-decreasing, then φ is convex. 4. Show that x → ex is convex on R. 5. Show that x → − ln(x) is convex on ]0, +∞[. Tutorial 8: Jensen inequality 16 Definition 70 we say that a finite measure space (Ω, F , P ) is a probability space, if and only if P (Ω) = 1. Definition 71 Let (Ω, F , P ) be a probability space, and (S, Σ) be a measurable space. We call random variable w.r. to (S, Σ), any measurable map X : (Ω, F ) → (S, Σ). Definition 72 Let (Ω, F , P ) be a probability space. Let X be a nonnegative random variable, or an element of L1 (Ω, F , P ). We call C expectation of X , denoted E [X ], the integral: E [X ] = XdP Ω Tutorial 8: Jensen inequality 17 ¯ Exercise 15. Let a, b ∈ R, a < b and φ : ]a, b[→ R be a convex map. Let (Ω, F , P ) be a probability space and X ∈ L1 (Ω, F , P ) be such R that X (Ω) ⊆]a, b[. 1. Show that φ ◦ X : (Ω, F ) → (R, B (R)) is measurable. 2. Show that φ ◦ X ∈ L1 (Ω, F , P ), if and only if E [|φ ◦ X |] < +∞. R 3. Show that if E [X ] = a, then a ∈ R and X = a P -a.s. 4. Show that if E [X ] = b, then b ∈ R and X = b P -a.s. 5. Let m = E [X ]. Show that m ∈]a, b[. 6. Define: β= sup x∈]a,m[ φ(m) − φ(x) m−x Show that β ∈ R and that for all z ∈]m, b[, we have: β≤ φ(z ) − φ(m) z−m Tutorial 8: Jensen inequality 18 7. Show that for all x ∈]a, b[, we have φ(m) + β (x − m) ≤ φ(x). 8. Show that for all ω ∈ Ω, φ(m) + β (X (ω ) − m) ≤ φ(X (ω )). 9. Show that if φ ◦ X ∈ L1 (Ω, F , P ) then φ(m) ≤ E [φ ◦ X ]. R Theorem 40 (Jensen inequality) Let (Ω, F , P ) be a probability ¯ space. Let a, b ∈ R, a < b and φ : ]a, b[→ R be a convex map. Suppose that X ∈ L1 (Ω, F , P ) is such that X (Ω) ⊆]a, b[ and such R that φ ◦ X ∈ L1 (Ω, F , P ). Then: R φ(E [X ]) ≤ E [φ ◦ X ] Tutorial 9: Lp -spaces, p ∈ [1, +∞] 1 9. Lp-spaces, p ∈ [1, +∞] In the following, (Ω, F , µ) is a measure space. Exercise 1. Let f, g : (Ω, F ) → [0, +∞] be non-negative and measurable maps. Let p, q ∈ R+ , such that 1/p + 1/q = 1. 1. Show that 1 < p < +∞ and 1 < q < +∞. 2. For all α ∈]0, +∞[, we define φα : [0, +∞] → [0, +∞] by: φα (x) = xα +∞ if if x ∈ R+ x = +∞ Show that φα is a continuous map. 3. Define A = ( f p dµ)1/p , B = ( g q dµ)1/q and C = f gdµ. Explain why A, B and C are well defined elements of [0, +∞]. 4. Show that if A = 0 or B = 0 then C ≤ AB . 5. Show that if A = +∞ or B = +∞ then C ≤ AB . Tutorial 9: Lp -spaces, p ∈ [1, +∞] 2 6. We assume from now on that 0 < A < +∞ and 0 < B < +∞. Define F = f /A and G = g/B . Show that: F p dµ = Ω Gp dµ = 1 Ω 7. Let a, b ∈]0, +∞[. Show that: ln(a) + ln(b) ≤ ln 1p 1q a+ b p q and: 1p 1q a+ b p q Prove this last inequality for all a, b ∈ [0, +∞]. ab ≤ 8. Show that for all ω ∈ Ω, we have: F (ω )G(ω ) ≤ 1 1 (F (ω ))p + (G(ω ))q p q Tutorial 9: Lp -spaces, p ∈ [1, +∞] 3 9. Show that C ≤ AB . Theorem 41 (H¨lder’s inequality) Let (Ω, F , µ) be a measure o space and f, g : (Ω, F ) → [0, +∞] be two non-negative and measurable maps. Let p, q ∈ R+ be such that 1/p + 1/q = 1. Then: f gdµ ≤ Ω f p dµ 1 p Ω g q dµ 1 q Ω Theorem 42 (Cauchy-Schwarz’s inequality:first) Let (Ω, F , µ) be a measure space and f, g : (Ω, F ) → [0, +∞] be two non-negative and measurable maps. Then: f gdµ ≤ Ω f 2 dµ Ω 1 2 g 2 dµ Ω 1 2 Tutorial 9: Lp -spaces, p ∈ [1, +∞] 4 Exercise 2. Let f, g : (Ω, F ) → [0, +∞] be two non-negative and measurable maps. Let p ∈]1, +∞[. Define A = ( f p dµ)1/p and B = ( g p dµ)1/p and C = ( (f + g )p dµ)1/p . 1. Explain why A, B and C are well defined elements of [0, +∞]. 2. Show that for all a, b ∈]0, +∞[, we have: (a + b)p ≤ 2p−1 (ap + bp ) Prove this inequality for all a, b ∈ [0, +∞]. 3. Show that if A = +∞ or B = +∞ or C = 0 then C ≤ A + B . 4. Show that if A < +∞ and B < +∞ then C < +∞. 5. We assume from now that A, B ∈ [0, +∞[ and C ∈]0, +∞[. Show the existence of some q ∈ R+ such that 1/p + 1/q = 1. 6. Show that for all a, b ∈ [0, +∞], we have: (a + b)p = (a + b).(a + b)p−1 Tutorial 9: Lp -spaces, p ∈ [1, +∞] 5 7. Show that: p f.(f + g )p−1 dµ ≤ AC q Ω p g.(f + g )p−1 dµ ≤ BC q Ω 8. Show that: p (f + g )p dµ ≤ C q (A + B ) Ω 9. Show that C ≤ A + B . 10. Show that the inequality still holds if we assume that p = 1. Theorem 43 (Minkowski’s inequality) Let (Ω, F , µ) be a measure space and f, g : (Ω, F ) → [0, +∞] be two non-negative and measurable maps. Let p ∈ [1, +∞[. Then: (f + g )p dµ Ω 1 p ≤ f p dµ Ω 1 p g p dµ + Ω 1 p Tutorial 9: Lp -spaces, p ∈ [1, +∞] 6 Definition 73 The Lp -spaces, p ∈ [1, +∞[, on (Ω, F , µ), are: Lp (Ω, F , µ) = f : (Ω, F ) → (R, B (R)) measurable, R Lp (Ω, F , µ) = f : (Ω, F ) → (C, B (C)) measurable, C |f |p dµ < +∞ Ω |f |p dµ < +∞ Ω For all f ∈ Lp (Ω, F , µ), we put: C f p |f | dµ p = 1 p Ω Exercise 3. Let p ∈ [1, +∞[. Let f, g ∈ Lp (Ω, F , µ). C 1. Show that Lp (Ω, F , µ) = {f ∈ Lp (Ω, F , µ) , f (Ω) ⊆ R}. R C 2. Show that Lp (Ω, F , µ) is closed under R-linear combinations. R 3. Show that Lp (Ω, F , µ) is closed under C-linear combinations. C Tutorial 9: Lp -spaces, p ∈ [1, +∞] 4. Show that f + g 5. Show that f p p ≤f 7 p + g p. = 0 ⇔ f = 0 µ-a.s. 6. Show that for all α ∈ C, αf 7. Explain why (f, g ) → f − g p p = |α|. f p. is not a metric on Lp (Ω, F , µ) C Definition 74 For all f : (Ω, F ) → (C, B (C)) measurable, Let: f ∞ = inf {M ∈ R+ , |f | ≤ M µ-a.s.} The L∞ -spaces on a measure space (Ω, F , µ) are: L∞ (Ω, F , µ) = {f : (Ω, F ) → (R, B (R)) measurable, f R ∞ < +∞} L∞ (Ω, F , µ) C ∞ < +∞} = {f : (Ω, F ) → (C, B (C)) measurable, f Tutorial 9: Lp -spaces, p ∈ [1, +∞] 8 Exercise 4. Let f, g ∈ L∞ (Ω, F , µ). C 1. Show that L∞ (Ω, F , µ) = {f ∈ L∞ (Ω, F , µ) , f (Ω) ⊆ R}. R C 2. Show that |f | ≤ f 3. Show that f + g ∞ ∞ µ-a.s. ≤f ∞ +g ∞. 4. Show that L∞ (Ω, F , µ) is closed under R-linear combinations. R 5. Show that L∞ (Ω, F , µ) is closed under C-linear combinations. C 6. Show that f ∞ = 0 ⇔ f = 0 µ-a.s.. 7. Show that for all α ∈ C, αf 8. Explain why (f, g ) → f − g ∞ ∞ = |α|. f ∞. is not a metric on L∞ (Ω, F , µ) C Tutorial 9: Lp -spaces, p ∈ [1, +∞] 9 Definition 75 Let p ∈ [1, +∞]. Let K = R or C. For all > 0 and f ∈ Lp (Ω, F , µ), we define the so-called open ball in Lp (Ω, F , µ): K K B (f, ) = {g : g ∈ Lp (Ω, F , µ), f − g K p <} We call usual topology in Lp (Ω, F , µ), the set T defined by: K T = {U : U ⊆ Lp (Ω, F , µ), ∀f ∈ U, ∃ > 0, B (f, ) ⊆ U } K Note that if (f, g ) → f − g p was a metric, the usual topology in Lp (Ω, F , µ), would be nothing but the metric topology. K Exercise 5. Let p ∈ [1, +∞]. Suppose there exists N ∈ F with µ(N ) = 0 and N = ∅. Put f = 1N and g = 0 1. Show that f, g ∈ Lp (Ω, F , µ) and f = g . C 2. Show that any open set containing f also contains g . 3. Show that Lp (Ω, F , µ) and Lp (Ω, F , µ) are not Hausdorff. C R Tutorial 9: Lp -spaces, p ∈ [1, +∞] 10 Exercise 6. Show that the usual topology on Lp (Ω, F , µ) is induced R by the usual topology on Lp (Ω, F , µ), where p ∈ [1, +∞]. C Definition 76 Let (E, T ) be a topological space. A sequence (xn )n≥1 T in E is said to converge to x ∈ E , and we write xn → x, if and only if, for all U ∈ T such that x ∈ U , there exists n0 ≥ 1 such that: n ≥ n0 ⇒ xn ∈ U Lp When E = Lp (Ω, F , µ) or E = Lp (Ω, F , µ), we write xn → x. C R Exercise 7. Let (E, T ) be a topological space and E ⊆ E . Let T = T|E be the induced topology on E . Show that if (xn )n≥1 is a T T sequence in E and x ∈ E , then xn → x is equivalent to xn → x. Exercise 8. Let f, g, (fn )n≥1 be in Lp (Ω, F , µ) and p ∈ [1, +∞]. C 1. Recall what the notation fn → f means. Lp 2. Show that fn → f is equivalent to fn − f p → 0. Tutorial 9: Lp -spaces, p ∈ [1, +∞] Lp 11 Lp 3. Show that if fn → f and fn → g then f = g µ-a.s. Exercise 9. Let p ∈ [1, +∞]. Suppose there exists some N ∈ F such that µ(N ) = 0 and N = ∅. Find a sequence (fn )n≥1 in Lp (Ω, F , µ) C Lp Lp and f, g in Lp (Ω, F , µ), f = g such that fn → f and fn → g . C Definition 77 Let (fn )n≥1 be a sequence in Lp (Ω, F , µ), where C (Ω, F , µ) is a measure space and p ∈ [1, +∞]. We say that (fn )n≥1 is a cauchy sequence, if and only if, for all > 0, there exists n0 ≥ 1 such that: n, m ≥ n0 ⇒ fn − fm p ≤ Exercise 10. Let f, (fn )n≥1 be in Lp (Ω, F , µ) and p ∈ [1, +∞]. C Lp Show that if fn → f , then (fn )n≥1 is a cauchy sequence. Tutorial 9: Lp -spaces, p ∈ [1, +∞] 12 Exercise 11. Let p ∈ [1, +∞], and (fn )n≥1 be cauchy in Lp (Ω, F , µ). C 1. Show the existence of n1 ≥ 1 such that: n ≥ n1 ⇒ fn − fn1 p ≤ 1 2 2. Suppose we have found n1 < n2 < . . . < nk , k ≥ 1, such that: ∀j ∈ {1, . . . , k } , n ≥ nj ⇒ fn − fnj p ≤ 1 2j Show the existence of nk+1 , nk < nk+1 such that: n ≥ nk+1 ⇒ fn − fnk+1 p ≤ 1 2k+1 3. Show that there exists a subsequence (fnk )k≥1 of (fn )n≥1 with: +∞ fnk+1 − fnk k=1 p < +∞ Tutorial 9: Lp -spaces, p ∈ [1, +∞] 13 Exercise 12. Let p ∈ [1, +∞], and (fn )n≥1 be in Lp (Ω, F , µ), with: C +∞ fn+1 − fn p < +∞ n=1 We define: +∞ |fn+1 − fn | g= n=1 1. Show that g : (Ω, F ) → [0, +∞] is non-negative and measurable. 2. If p = +∞, show that g ≤ +∞ n=1 fn+1 − fn ∞ µ-a.s. 3. If p ∈ [1, +∞[, show that for all N ≥ 1, we have: +∞ N |fn+1 − fn | n=1 ≤ p fn+1 − fn n=1 p Tutorial 9: Lp -spaces, p ∈ [1, +∞] 14 4. If p ∈ [1, +∞[, show that: p g dµ Ω 1 p +∞ ≤ fn+1 − fn p n=1 5. Show that for p ∈ [1, +∞], we have g < +∞ µ-a.s. 6. Define A = {g < +∞}. Show that for all ω ∈ A, (fn (ω ))n≥1 is a cauchy sequence in C. We denote z (ω) its limit. 7. Define f : (Ω, F ) → (C, B (C)), by: f (ω) = z (ω) 0 if if ω∈A ω∈A Show that f is measurable and fn → f µ-a.s. 8. if p = +∞, show that for all n ≥ 1, |fn | ≤ |f1 | + g and conclude that f ∈ L∞ (Ω, F , µ). C Tutorial 9: Lp -spaces, p ∈ [1, +∞] 15 9. If p ∈ [1, +∞[, show the existence of n0 ≥ 1, such that: n ≥ n0 ⇒ |fn − fn0 |p dµ ≤ 1 Ω Deduce from Fatou’s lemma that f − fn0 ∈ Lp (Ω, F , µ). C 10. Show that for p ∈ [1, +∞], f ∈ Lp (Ω, F , µ). C 11. Suppose that fn ∈ Lp (Ω, F , µ), for all n ≥ 1. Show the exisR tence of f ∈ Lp (Ω, F , µ), such that fn → f µ-a.s. R Exercise 13. Let p ∈ [1, +∞], and (fn )n≥1 be in Lp (Ω, F , µ), with: C +∞ fn+1 − fn p < +∞ n=1 1. Does there exist f ∈ Lp (Ω, F , µ) such that fn → f µ-a.s. C Tutorial 9: Lp -spaces, p ∈ [1, +∞] 16 2. Suppose p = +∞. Show that for all n < m, we have: m |fm+1 − fn | ≤ fk+1 − fk ∞ µ-a.s. k =n 3. Suppose p = +∞. Show that for all n ≥ 1, we have: +∞ f − fn ∞ ≤ fk+1 − fk ∞ k =n 4. Suppose p ∈ [1, +∞[. Show that for all n < m, we have: |fm+1 − fn | dµ p 1 p Ω m ≤ fk+1 − fk p k =n 5. Suppose p ∈ [1, +∞[. Show that for all n ≥ 1, we have: +∞ f − fn p ≤ fk+1 − fk k =n p Tutorial 9: Lp -spaces, p ∈ [1, +∞] 17 Lp 6. Show that for p ∈ [1, +∞], we also have fn → f . Lp 7. Suppose conversely that g ∈ Lp (Ω, F , µ) is such that fn → g . C Show that f = g µ-a.s.. Conclude that fn → g µ-a.s.. Theorem 44 Let (Ω, F , µ) be a measure space. Let p ∈ [1, +∞], and (fn )n≥1 be a sequence in Lp (Ω, F , µ) such that: C +∞ fn+1 − fn p < +∞ n=1 Then, there exists f ∈ Lp (Ω, F , µ) such that fn → f µ-a.s. Moreover, C Lp for all g ∈ Lp (Ω, F , µ), the convergence fn → g µ-a.s. and fn → g C are equivalent. Tutorial 9: Lp -spaces, p ∈ [1, +∞] 18 Lp Exercise 14. Let f, (fn )n≥1 be in Lp (Ω, F , µ) such that fn → f , C where p ∈ [1, +∞]. 1. Show that there exists a sub-sequence (fnk )k≥1 of (fn )n≥1 , with: +∞ fnk+1 − fnk p < +∞ k=1 2. Show that there exists g ∈ Lp (Ω, F , µ) such that fnk → g µ-a.s. C Lp 3. Show that fnk → g and g = f µ-a.s. 4. Conclude with the following: Theorem 45 Let (fn )n≥1 be in Lp (Ω, F , µ) and f ∈ Lp (Ω, F , µ) C C Lp such that fn → f , where p ∈ [1, +∞]. Then, we can extract a subsequence (fnk )k≥1 of (fn )n≥1 such that fnk → f µ-a.s. Tutorial 9: Lp -spaces, p ∈ [1, +∞] 19 Exercise 15. Prove the last theorem for Lp (Ω, F , µ). R Exercise 16. Let p ∈ [1, +∞], and (fn )n≥1 be cauchy in Lp (Ω, F , µ). C 1. Show that there exists a subsequence (fnk )k≥1 of (fn )n≥1 and Lp f belonging to Lp (Ω, F , µ), such that fnk → f . C Lp 2. Using the fact that (fn )n≥1 is cauchy, show that fn → f . Theorem 46 Let p ∈ [1, +∞]. Let (fn )n≥1 be a cauchy sequence in Lp Lp (Ω, F , µ). Then, there exists f ∈ Lp (Ω, F , µ) such that fn → f . C C Exercise 17. Prove the last theorem for Lp (Ω, F , µ). R Tutorial 10: Bounded Linear Functionals in L2 1 10. Bounded Linear Functionals in L2 In the following, (Ω, F , µ) is a measure space. Definition 78 Let (xn )n≥1 be a sequence in an arbitrary set. We call subsequence of (xn )n≥1 , any sequence of the form (xφ(n) )n≥1 , where φ : N∗ → N∗ is a strictly increasing map. Exercise 1. Let (E, d) be a metric space, with metric topology T . Let (xn )n≥1 be a sequence in E . For all n ≥ 1, let Fn be the closure of the set {xk : k ≥ n}. T 1. Show that for all x ∈ E , xn → x is equivalent to: ∀ > 0 , ∃n0 ≥ 1 , n ≥ n0 ⇒ d(xn , x) ≤ 2. Show that (Fn )n≥1 is a decreasing sequence of closed sets in E . c 3. Show that if Fn ↓ ∅, then (Fn )n≥1 is an open covering of E . Tutorial 10: Bounded Linear Functionals in L2 2 ∞ 4. Show that if (E, T ) is compact then ∩+=1 Fn = ∅. n 5. Show that if (E, T ) is compact, there exists x ∈ E such that for all n ≥ 1 and > 0, we have B (x, ) ∩ {xk , k ≥ n} = ∅. 6. By induction, construct a subsequence (xnp )p≥1 of (xn )n≥1 such that xnp ∈ B (x, 1/p) for all p ≥ 1. 7. Conclude that if (E, T ) is compact, any sequence (xn )n≥1 in E has a convergent subsequence. Exercise 2. Let (E, d) be a metric space, with metric topology T . We assume that any sequence (xn )n≥1 in E has a convergent subsequence. Let (Vi )i∈I be an open covering of E . For x ∈ E , let: r(x) = sup{r > 0 : B (x, r) ⊆ Vi , for some i ∈ I } 1. Show that ∀x ∈ E , ∃i ∈ I , ∃r > 0, such that B (x, r) ⊆ Vi . 2. Show that ∀x ∈ E , r(x) > 0. Tutorial 10: Bounded Linear Functionals in L2 3 Exercise 3. Further to ex. (2), suppose inf x∈E r(x) = 0. 1. Show that for all n ≥ 1, there is xn ∈ E such that r(xn ) < 1/n. 2. Extract a subsequence (xnk )k≥1 of (xn )n≥1 converging to some x∗ ∈ E . Let r∗ > 0 and i ∈ I be such that B (x∗ , r∗ ) ⊆ Vi . Show that we can find some k0 ≥ 1, such that d(x∗ , xnk0 ) < r∗ /2 and r(xnk0 ) ≤ r∗ /4. 3. Show that d(x∗ , xnk0 ) < r∗ /2 implies that B (xnk0 , r∗ /2) ⊆ Vi . Show that this contradicts r(xnk0 ) ≤ r∗ /4, and conclude that inf x∈E r(x) > 0. Exercise 4. Further to ex. (3), Let r0 with 0 < r0 < inf x∈E r(x). Suppose that E cannot be covered by a finite number of open balls with radius r0 . 1. Show the existence of a sequence (xn )n≥1 in E , such that for all n ≥ 1, xn+1 ∈ B (x1 , r0 ) ∪ . . . ∪ B (xn , r0 ). Tutorial 10: Bounded Linear Functionals in L2 4 2. Show that for all n > m we have d(xn , xm ) ≥ r0 . 3. Show that (xn )n≥1 cannot have a convergent subsequence. 4. Conclude that there exists a finite subset {x1 , . . . , xn } of E such that E = B (x1 , r0 ) ∪ . . . ∪ B (xn , r0 ). 5. Show that for all x ∈ E , we have B (x, r0 ) ⊆ Vi for some i ∈ I . 6. Conclude that (E, T ) is compact. 7. Prove the following: Theorem 47 Let (E, T ) be a metrizable topological space. Then (E, T ) is compact, if and only if for every sequence (xn )n≥1 in E , there exists a subsequence (xnk )k≥1 of (xn )n≥1 , and some x ∈ E , T such that xnk → x. Tutorial 10: Bounded Linear Functionals in L2 5 Exercise 5. Let a, b ∈ R , a < b and (xn )n≥1 be a sequence in ]a, b[. 1. Show that (xn )n≥1 has a convergent subsequence. 2. Can we conclude that ]a, b[ is a compact subset of R? Exercise 6. Let E = [−M, M ] × . . . × [−M, M ] ⊆ Rn , where n ≥ 1 and M ∈ R+ . Let TRn be the usual product topology on Rn , and TE = (TRn )|E be the induced topology on E . T E 1. Let (xp )p≥1 be a sequence in E . Let x ∈ E . Show that xp → x T n R is equivalent to xp → x. 2. Propose a metric on Rn , inducing the topology TRn . 3. Let (xp )p≥1 be a sequence in Rn . Let x ∈ Rn . Show that T n T R R xp → x if and only if, xi → xi for all i ∈ Nn . p Tutorial 10: Bounded Linear Functionals in L2 6 Exercise 7. Further to ex. (6), suppose (xp )p≥1 is a sequence in E . 1. Show the existence of a subsequence (xφ(p) )p≥1 of (xp )p≥1 , such that x1 (p) φ T[−M,M ] → x1 for some x1 ∈ [−M, M ]. T R 2. Explain why the above convergence is equivalent to x1 (p) → x1 . φ 3. Suppose that 1 ≤ k ≤ n − 1 and (yp )p≥1 = (xφ(p) )p≥1 is a subsequence of (xp )p≥1 such that: T R ∀j = 1, . . . , k , xj (p) → xj for some xj ∈ [−M, M ] φ Show the existence of a subsequence (yψ(p) )p≥1 of (yp )p≥1 such T R k that yψ+1 → xk+1 for some xk+1 ∈ [−M, M ]. (p) 4. Show that φ ◦ ψ : N∗ → N∗ is strictly increasing. 5. Show that (xφ◦ψ(p) )p≥1 is a subsequence of (xp )p≥1 such that: T R ∀j = 1, . . . , k + 1 , xj ◦ψ(p) → xj ∈ [−M, M ] φ Tutorial 10: Bounded Linear Functionals in L2 7 6. Show the existence of a subsequence (xφ(p) )p≥1 of (xp )p≥1 , and T E x ∈ E , such that xφ(p) → x 7. Show that (E, TE ) is a compact topological space. Exercise 8. Let A be a closed subset of Rn , n ≥ 1, which is bounded with respect to the usual metric of Rn . 1. Show that A ⊆ E = [−M, M ]×. . .×[−M, M ], for some M ∈ R+ . 2. Show from E \ A = E ∩ Ac that A is closed in E . 3. Show (A, (TRn )|A ) is a compact topological space. 4. Conversely, let A is a compact subset of Rn . Show that A is closed and bounded. Theorem 48 A subset of Rn , n ≥ 1, is compact if and only if it is closed and bounded (with respect to the usual metric). Tutorial 10: Bounded Linear Functionals in L2 8 Exercise 9. Let n ≥ 1. Consider the map: φ: Cn → R2n (a1 + ib1 , . . . , an + ibn ) → (a1 , b1 , . . . , an , bn ) 1. Recall the expressions of the usual metrics dCn and dR2n of Cn and R2n respectively. 2. Show that for all z, z ∈ Cn , dCn (z, z ) = dR2n (φ(z ), φ(z )). 3. Show that φ is a homeomorphism from Cn to R2n . 4. Show that a subset K of Cn is compact, if and only if φ(K ) is a compact subset of R2n . 5. Show that K is closed, if and only if φ(K ) is closed. 6. Show that K is bounded, if and only if φ(K ) is bounded. 7. Show that a subset K of Cn is compact, if and only if it is closed and bounded (with respect to the usual metric). Tutorial 10: Bounded Linear Functionals in L2 9 Definition 79 Let (E, d) be a metric space. A sequence (xn )n≥1 in E , is said to be a cauchy sequence (relative to the metric d), if and only if, for all > 0, there exists n0 ≥ 1 such that: n, m ≥ n0 ⇒ d(xn , xm ) ≤ Definition 80 We say that a metric space (E, d) is complete, if and only if, for all (xn )n≥1 cauchy sequence in E , there exists x ∈ E such that (xn )n≥1 converges to x. Exercise 10. 1. Explain why strictly speaking, given p ∈ [1, +∞], definition (77) of Cauchy sequences in Lp (Ω, F , µ) is not a covered by definiC tion (79). 2. Explain why Lp (Ω, F , µ) is not a complete metric space, despite C theorem (46) and definition (80). Tutorial 10: Bounded Linear Functionals in L2 10 Exercise 11. Let (zk )k≥1 be a Cauchy sequence in Cn , n ≥ 1, with respect to the usual metric d(z, z ) = z − z , where: n |zi |2 z= i=1 1. Show that the sequence (zk )k≥1 is bounded, i.e. that there exists M ∈ R+ such that zk ≤ M , for all k ≥ 1. 2. Define B = {z ∈ Cn , z ≤ M }. Show that δ (B ) < +∞, and that B is closed in Cn . 3. Show the existence of a subsequence (zkp )p≥1 of (zk )k≥1 such T n C that zkp → z for some z ∈ B . 4. Show that for all > 0, there exists p0 ≥ 1 and n0 ≥ 1 such that d(z, zkp0 ) ≤ /2 and: k ≥ n0 ⇒ d(zk , zkp0 ) ≤ /2 Tutorial 10: Bounded Linear Functionals in L2 T 11 n C 5. Show that zk → z . 6. Conclude that Cn is complete with respect to its usual metric. 7. For which theorem of Tutorial 9 was the completeness of C used? T n C Exercise 12. Let (xk )k≥1 be a sequence in Rn such that xk → z , for some z ∈ Cn . 1. Show that z ∈ Rn . 2. Show that Rn is complete with respect to its usual metric. Theorem 49 For all n ≥ 1, Cn and Rn are complete with respect to their usual metrics. Tutorial 10: Bounded Linear Functionals in L2 12 Exercise 13. Let (E, d) be a metric space, with metric topology T . ¯ Let F ⊆ E , and F denote the closure of F . ¯ 1. Explain why, for all x ∈ F and n ≥ 1, we have F ∩ B (x, 1/n) = ∅. ¯ 2. Show that for all x ∈ F , there exists a sequence (xn )n≥1 in F , T such that xn → x. 3. Show conversely that if there is a sequence (xn )n≥1 in F with T ¯ xn → x, then x ∈ F . 4. Show that F is closed if and only if for all sequence (xn )n≥1 in T F such that xn → x for some x ∈ E , we have x ∈ F . 5. Explain why (F, T|F ) is metrizable. 6. Show that if F is complete with respect to the metric d|F ×F , then F is closed in E . Tutorial 10: Bounded Linear Functionals in L2 13 ¯ 7. Let dR be a metric on R, inducing the usual topology TR . Show ¯ ¯ that d = (dR )|R×R is a metric on R, inducing the topology TR . ¯ 8. Find a metric on [−1, 1] which induces its usual topology. 9. Show that {−1, 1} is not open in [−1, 1]. ¯ 10. Show that {−∞, +∞} is not open in R. ¯ 11. Show that R is not closed in R. 12. Let dR be the usual metric of R. Show that d = (dR )|R×R ¯ and dR induce the same topology on R, but that however, R is complete with respect to dR , whereas it cannot be complete with respect to d . Tutorial 10: Bounded Linear Functionals in L2 14 Definition 81 Let H be a K-vector space, where K = R or C. We call inner-product on H, any map ·, · : H × H → K with the following properties: (i) (ii) (iii) (iv ) (v ) ∀x, y ∈ H , x, y = y , x ∀x, y, z ∈ H , x + z, y = x, y + z , y ∀x, y ∈ H, ∀α ∈ K , αx, y = α x, y ∀x ∈ H , x, x ≥ 0 ∀x ∈ H , ( x, x = 0 ⇔ x = 0) where for all z ∈ C, z denotes the complex conjugate of z . For all ¯ x ∈ H, we call norm of x, denoted x , the number defined by: x= x, x Exercise 14. Let ·, · be an inner-product on a K-vector space H. 1. Show that for all y ∈ H, the map x → x, y is linear. Tutorial 10: Bounded Linear Functionals in L2 15 2. Show that for all x ∈ H, the map y → x, y is linear if K = R, and conjugate-linear if K = C. Exercise 15. Let ·, · be an inner-product on a K-vector space H. Let x, y ∈ H. Let A = x 2 , B = | x, y | and C = y 2 . let α ∈ K be such that |α| = 1 and: B = α x, y 1. Show that A, B, C ∈ R+ . 2. For all t ∈ R, show that x − tαy, x − tαy = A − 2tB + t2 C . 3. Show that if C = 0 then B 2 ≤ AC . 4. Suppose that C = 0. Show that P (t) = A − 2tB + t2 C has a minimal value which is in R+ , and conclude that B 2 ≤ AC . 5. Conclude with the following: Tutorial 10: Bounded Linear Functionals in L2 16 Theorem 50 (Cauchy-Schwarz’s inequality:second) Let H be a K-vector space, where K = R or C, and ·, · be an inner-product on H. Then, for all x, y ∈ H, we have: | x, y | ≤ x . y Exercise 16. For all f, g ∈ L2 (Ω, F , µ), we define: C f, g = f gdµ ¯ Ω 1. Use the first cauchy-schwarz inequality (42) to prove that for all ¯ f, g ∈ L2 (Ω, F , µ), we have f g ∈ L1 (Ω, F , µ). Conclude that C C f , g is a well-defined complex number. 2. Show that for all f ∈ L2 (Ω, F , µ), we have f C 2 = f, f . 3. Make another use of the first cauchy-schwarz inequality to show that for all f, g ∈ L2 (Ω, F , µ), we have: C | f, g | ≤ f 2. g 2 (1) Tutorial 10: Bounded Linear Functionals in L2 17 4. Go through definition (81), and indicate which of the properties (i) − (v ) fails to be satisfied by ·, · . Conclude that ·, · is not an inner-product on L2 (Ω, F , µ), and therefore that inC equality (*) is not a particular case of the second cauchy-schwarz inequality (50). 5. Let f, g ∈ L2 (Ω, F , µ). By considering (|f |+t|g |)2 dµ for t ∈ R, C imitate the proof of the second cauchy-schwarz inequality to show that: |f g |dµ ≤ Ω |f |2 dµ 1 2 Ω 1 2 |g |2 dµ Ω 6. Let f, g : (Ω, F ) → [0, +∞] non-negative and measurable. Show that if f 2 dµ and g 2 dµ are finite, then f and g are µ-almost surely equal to elements of L2 (Ω, F , µ). Deduce from 5. a new C proof of the first Cauchy-Schwarz inequality: f gdµ ≤ Ω f 2 dµ Ω 1 2 g 2 dµ Ω 1 2 Tutorial 10: Bounded Linear Functionals in L2 18 Exercise 17. Let ·, · be an inner product on a K-vector space H. 1. Show that for all x, y ∈ H, we have: x+y 2 =x 2 +y 2 + x, y + x, y 2. Using the second cauchy-schwarz inequality (50), show that: x+y ≤ x + y 3. Show that d ·,· (x, y ) = x − y defines a metric on H. Definition 82 Let H be a K-vector space, where K = R or C, and ·, · be an inner-product on H. We call norm topology on H, denoted T ·,· , the metric topology associated with d ·,· (x, y ) = x − y . Definition 83 We call hilbert space (over K), where K = R or C, any ordered pair (H, ·, · ), where H is a K-vector space, and ·, · is an inner product on H for which the metric space (H, d ·,· ) is complete, where d ·,· (x, y ) = x − y . Tutorial 10: Bounded Linear Functionals in L2 19 Exercise 18. Let (H, ·, · ) be a hilbert space over K and let M be a closed linear subspace of H, (closed with respect to the norm topology T ·,· ). Define [·, ·] = ·, · |M×M . 1. Show that [·, ·] is an inner-product on the K-vector space M. 2. With obvious notations, show that d[,·] = (d 3. Deduce that T[·,·] = (T ·,· ·,· )|M×M . )|M . Exercise 19. Further to ex. (18), Let (xn )n≥1 be a cauchy sequence in M, with respect to the metric d[·,·] . 1. Show that (xn )n≥1 is a cauchy sequence in H. T ·,· 2. Explain why there exists x ∈ H such that xn → x. 3. Explain why x ∈ M. T[·,·] 4. Explain why we also have xn → x. Tutorial 10: Bounded Linear Functionals in L2 5. Explain why (M, ·, · |M×M ) 20 is a hilbert space over K. Exercise 20. For all z, z ∈ Cn , n ≥ 1, we define: n z, z = zi zi ¯ i=1 1. Show that ·, · is an inner-product on Cn . 2. Show that the metric d ·,· is equal to the usual metric of Cn . 3. Conclude that (Cn , ·, · ) is a hilbert space over C. 4. Show that Rn is a closed subset of Cn . 5. Show however that Rn is not a linear subspace of Cn . 6. Show that (Rn , ·, · | R n ×R n ) is a hilbert space over R. Tutorial 10: Bounded Linear Functionals in L2 21 Definition 84 Let K = R or C. The usual inner-product in Kn , denoted ·, · , is defined as: n ∀x, y ∈ Kn , x, y = xi yi ¯ i=1 Theorem 51 The spaces Cn and Rn , n ≥ 1, together with their usual inner-products, are hilbert spaces over C and R respectively. Definition 85 Let H be a K-vector space, where K = R or C. Let C ⊆ H. We say that C is a convex subset or H, if and only if, for all x, y ∈ C and t ∈ [0, 1], we have tx + (1 − t)y ∈ C . Tutorial 10: Bounded Linear Functionals in L2 22 Exercise 21. Let (H, ·, · ) be a hilbert space over K. Let C ⊆ H be a non-empty closed convex subset of H. Let x0 ∈ H. Define: δmin = inf { x − x0 : x ∈ C} 1. Show the existence of a sequence (xn )n≥1 in C such that xn − x0 → δmin . 2. Show that for all x, y ∈ H, we have: x−y 2 =2 x 2 +2 y 2 −4 x+y 2 2 3. Explain why for all n, m ≥ 1, we have: δmin ≤ xn + xm − x0 2 4. Show that for all n, m ≥ 1, we have: xn − xm 2 ≤ 2 xn − x0 2 + 2 xm − x0 2 2 − 4δmin Tutorial 10: Bounded Linear Functionals in L2 23 T ·,· 5. Show the existence of some x∗ ∈ H, such that xn → x∗ . 6. Explain why x∗ ∈ C 7. Show that for all x, y ∈ H, we have | x − y | ≤ x − y . 8. Show that xn − x0 → x∗ − x0 . 9. Conclude that we have found x∗ ∈ C such that: x∗ − x0 = inf { x − x0 : x ∈ C} 10. Let y ∗ be another element of C with such property. Show that: x∗ − y ∗ 2 ≤ 2 x∗ − x0 11. Conclude that x∗ = y ∗ . 2 + 2 y ∗ − x0 2 2 − 4δmin Tutorial 10: Bounded Linear Functionals in L2 24 Theorem 52 Let (H, ·, · ) be a hilbert space over K, where K = R or C. Let C be a non-empty, closed and convex subset of H. For all x0 ∈ H, there exists a unique x∗ ∈ C such that: x∗ − x0 = inf { x − x0 : x ∈ C} Definition 86 Let (H, ·, · ) be a hilbert space over K, where K = R or C. Let G ⊆ H. We call orthogonal of G , the subset of H denoted G ⊥ and defined by: G⊥ = { x ∈ H : x, y = 0 , ∀y ∈ G } Exercise 22. Let (H, ·, · ) be a hilbert space over K and G ⊆ H. 1. Show that G ⊥ is a linear subspace of H, even if G isn’t. 2. Show that φy : H → K defined by φy (x) = x, y is continuous. 3. Show that G ⊥ = ∩y∈G φ−1 ({0}). y Tutorial 10: Bounded Linear Functionals in L2 25 4. Show that G ⊥ is a closed subset of H, even if G isn’t. 5. Show that ∅⊥ = {0}⊥ = H. 6. Show that H⊥ = {0}. Exercise 23. Let (H, ·, · ) be a hilbert space over K. Let M be a closed linear subspace of H, and x0 ∈ H. 1. Explain why there exists x∗ ∈ M such that: x∗ − x0 = inf { x − x0 : x∈M} 2. Define y ∗ = x0 − x∗ ∈ H. Show that for all y ∈ M and α ∈ K: y∗ 2 ≤ y ∗ − αy 2 3. Show that for all y ∈ M and α ∈ K, we have: 0 ≤ −α y , y ∗ − α y , y ∗ + |α|2 . y 2 Tutorial 10: Bounded Linear Functionals in L2 26 4. For all y ∈ M \ {0}, taking α = y , y ∗ / y 2 , show that: 0≤− | y, y ∗ |2 y2 5. Conclude that x∗ ∈ M, y ∗ ∈ M⊥ and x0 = x∗ + y ∗ . 6. Show that M ∩ M⊥ = {0} 7. Show that x∗ ∈ M and y ∗ ∈ M⊥ with x0 = x∗ + y ∗ , are unique. Theorem 53 Let (H, ·, · ) be a hilbert space over K, where K = R or C. Let M be a closed linear subspace of H. Then, for all x0 ∈ H, there is a unique decomposition: x0 = x∗ + y ∗ where x∗ ∈ M and y ∗ ∈ M⊥ . Tutorial 10: Bounded Linear Functionals in L2 27 Definition 87 Let H be a K-vector space, where K = R or C. We call linear functional, any map λ : H → K, such that for all x, y ∈ H and α ∈ K: λ(x + αy ) = λ(x) + αλ(y ) Exercise 24. Let λ be a linear functional on a K-hilbert1 space H. 1. Suppose that λ is continuous at some point x0 ∈ H. Show the existence of η > 0 such that: ∀x ∈ H , x − x0 ≤ η ⇒ |λ(x) − λ(x0 )| ≤ 1 Show that for all x ∈ H with x = 0, we have |λ(ηx/ x )| ≤ 1. 2. Show that if λ is continuous at x0 , there exits M ∈ R+ , with: ∀x ∈ H , |λ(x)| ≤ M x 1 Norm vector spaces are introduced later in these tutorials. (2) Tutorial 10: Bounded Linear Functionals in L2 28 3. Show conversely that if (2) holds, λ is continuous everywhere. Definition 88 Let (H, ·, · ) be a hilbert2 space over K = R or C. Let λ be a linear functional on H. Then, the following are equivalent: (i) (ii) λ : (H, T ·,· ) → (K, TK ) is continuous ∃M ∈ R+ , ∀x ∈ H , |λ(x)| ≤ M. x In which case, we say that λ is a bounded linear functional. Exercise 25. Let (H, ·, · ) be a hilbert space over K. Let λ be a bounded linear functional on H, such that λ(x) = 0 for some x ∈ H, and define M = λ−1 ({0}). 1. Show the existence of x0 ∈ H, such that x0 ∈ M. 2. Show the existence of x∗ ∈ M and y ∗ ∈ M⊥ with x0 = x∗ + y ∗ . 2 Norm vector spaces are introduced later in these tutorials. Tutorial 10: Bounded Linear Functionals in L2 3. Deduce the existence of some z ∈ M⊥ such that z = 1. 4. Show that for all α ∈ K \ {0} and x ∈ H, we have: λ(x) z , αz = λ(x) α ¯ 5. Show that in order to have: ∀x ∈ H , λ(x) = x, αz it is sufficient to choose α ∈ K \ {0} such that: ∀x ∈ H , λ(x)z −x∈M α ¯ 6. Show the existence of y ∈ H such that: ∀x ∈ H , λ(x) = x, y 7. Show the uniqueness of such y ∈ H. 29 Tutorial 10: Bounded Linear Functionals in L2 30 Theorem 54 Let (H, ·, · ) be a hilbert space over K, where K = R or C. Let λ be a bounded linear functional on H. Then, there exists a unique y ∈ H such that: ∀x ∈ H , λ(x) = x, y . Definition 89 Let K = R or C. We call K -vector space, any set H, together with operators ⊕ and ⊗ for which there exits an element 0H ∈ H such that for all x, y, z ∈ H and α, β ∈ K, we have: (i) 0H ⊕ x = x (ii) (iii) ∃(−x) ∈ H , (−x) ⊕ x = 0H x ⊕ (y ⊕ z ) = (x ⊕ y ) ⊕ z (iv ) (v ) x⊕y =y⊕x 1⊗x=x (vi) α ⊗ (β ⊗ x) = (αβ ) ⊗ x (vii) (viii) (α + β ) ⊗ x = (α ⊗ x) ⊕ (β ⊗ x) α ⊗ (x ⊕ y ) = (α ⊗ x) ⊕ (α ⊗ y ) Tutorial 10: Bounded Linear Functionals in L2 31 Exercise 26. For all f ∈ L2 (Ω, F , µ), define: K H = { [f ] : f ∈ L2 (Ω, F , µ) } K where [f ] = {g ∈ L2 (Ω, F , µ) : g = f, µ-a.s.}. Let 0H = [0], and for K all [f ], [g ] ∈ H, and α ∈ K, we define: [f ] ⊕ [g ] = [f + g ] α ⊗ [f ] = [αf ] We assume f, f , g and g are elements of L2 (Ω, F , µ). K 1. Show that for f = g µ-a.s. is equivalent to [f ] = [g ]. 2. Show that if [f ] = [f ] and [g ] = [g ], then [f + g ] = [f + g ]. 3. Conclude that ⊕ is well-defined. 4. Show that ⊗ is also well-defined. 5. Show that (H, ⊕, ⊗) is a K-vector space. Tutorial 10: Bounded Linear Functionals in L2 32 Exercise 27. Further to ex. (26), we define for all [f ], [g ] ∈ H: [f ], [g ] H = f gdµ ¯ Ω 1. Show that ·, · H is well-defined. 2. Show that ·, · H is an inner-product on H. 3. Show that (H, ·, · 4. Why is f , g = H) Ω is a hilbert space over K. f g dµ not an inner-product on L2 (Ω, F , µ)? ¯ K Exercise 28. Further to ex. (27), Let λ : L2 (Ω, F , µ) → K be a K continuous linear functional3 . Define Λ : H → K by Λ([f ]) = λ(f ). 3 As defined in these tutorials, L2 (Ω, F , µ) is not a hilbert space (not even a K norm vector space). However, both L2 (Ω, F , µ) and K have natural topologies K and it is therefore meaningful to speak of continuous linear functional. Note however that we are slightly outside the framework of definition (88). Tutorial 10: Bounded Linear Functionals in L2 33 1. Show the existence of M ∈ R+ such that: ∀f ∈ L2 (Ω, F , µ) , |λ(f )| ≤ M. f K 2 2. Show that if [f ] = [g ] then λ(f ) = λ(g ). 3. Show that Λ is a well defined bounded linear functional on H. 4. Conclude with the following: Theorem 55 Let λ : L2 (Ω, F , µ) → K be a continuous linear funcK tional, where K = R or C. Then, there exists g ∈ L2 (Ω, F , µ) such K that: f gdµ ¯ ∀f ∈ L2 (Ω, F , µ) , λ(f ) = K Ω Tutorial 11: Complex Measures 1 11. Complex Measures In the following, (Ω, F ) denotes an arbitrary measurable space. Definition 90 Let (an )n≥1 be a sequence of complex numbers. We say that (an )n≥1 has the permutation property if and only if, for +∞ all bijections σ : N∗ → N∗ , the series k=1 aσ(k) converges in C1 Exercise 1. Let (an )n≥1 be a sequence of complex numbers. 1. Show that if (an )n≥1 has the permutation property, then the same is true of (Re(an ))n≥1 and (Im(an ))n≥1 . 2. Suppose an ∈ R for all n ≥ 1. Show that if +∞ +∞ |ak | = +∞ ⇒ k=1 1 which excludes ±∞ as limit. +∞ a+ = k k=1 k=1 +∞ k=1 ak converges: a− = + ∞ k Tutorial 11: Complex Measures 2 Exercise 2. Let (an )n≥1 be a sequence in R, such that the series +∞ +∞ k=1 ak converges, and k=1 |ak | = +∞. Let A > 0. We define: N + = { k ≥ 1 : ak ≥ 0 } , N − = { k ≥ 1 : ak < 0 } 1. Show that N + and N − are infinite. 2. Let φ+ : N∗ → N + and φ− : N∗ → N − be two bijections. Show the existence of k1 ≥ 1 such that: k1 aφ+ (k ) ≥ A k=1 3. Show the existence of an increasing sequence (kp )p≥1 such that: kp aφ+ (k ) ≥ A k=kp−1 +1 for all p ≥ 1, where k0 = 0. Tutorial 11: Complex Measures 3 4. Consider the permutation σ : N∗ → N∗ defined informally by: (φ− (1), φ+ (1), . . . , φ+ (k1 ), φ− (2), φ+ (k1 + 1), . . . , φ+ (k2 ), . . .) ∗ representing (σ (1), σ (2), . . .). More specifically, define k0 = 0 ∗ ∗ and kp = kp + p for all p ≥ 1. For all n ∈ N and p ≥ 1 with: 2 ∗ ∗ kp−1 < n ≤ kp (1) we define: σ (n) = if φ− (p) φ+ (n − p) if ∗ n = kp−1 + 1 ∗ n > kp−1 + 1 (2) Show that σ : N∗ → N∗ is indeed a bijection. 5. Show that if +∞ k=1 aσ(k) converges, there is N ≥ 1, such that: n+p n≥N , p≥1 ⇒ aσ (k ) < A k=n+1 2 Given an integer n ≥ 1, there exists a unique p ≥ 1 such that (1) holds. Tutorial 11: Complex Measures 4 6. Explain why (an )n≥1 cannot have the permutation property. 7. Prove the following theorem: Theorem 56 Let (an )n≥1 be a sequence of complex numbers such +∞ that for all bijections σ : N∗ → N∗ , the series k=1 aσ(k) converges. +∞ Then, the series k=1 ak converges absolutely, i.e. +∞ |ak | < +∞ k=1 Definition 91 Let (Ω, F ) be a measurable space and E ∈ F . We call measurable partition of E , any sequence (En )n≥1 of pairwise disjoint elements of F , such that E = n≥1 En . Tutorial 11: Complex Measures 5 Definition 92 We call complex measure on a measurable space (Ω, F ) any map µ : F → C, such that for all E ∈ F and (En )n≥1 +∞ measurable partition of E , the series n=1 µ(En ) converges to µ(E ). The set of all complex measures on (Ω, F ) is denoted M 1 (Ω, F ). Definition 93 We call signed measure on a measurable space (Ω, F ), any complex measure on (Ω, F ) with values in R.3 Exercise 3. 1. Show that a measure on (Ω, F ) may not be a complex measure. 2. Show that for all µ ∈ M 1 (Ω, F ) , µ(∅) = 0. 3. Show that a finite measure on (Ω, F ) is a complex measure with values in R+ , and conversely. 3 In these tutorials, signed measure may not have values in {−∞, +∞}. Tutorial 11: Complex Measures 6 4. Let µ ∈ M 1 (Ω, F ). Let E ∈ F and (En )n≥1 be a measurable partition of E . Show that: +∞ |µ(En )| < +∞ n=1 5. Let µ be a measure on (Ω, F ) and f ∈ L1 (Ω, F , µ). Define: C ∀E ∈ F , ν (E ) = f dµ E Show that ν is a complex measure on (Ω, F ). Definition 94 Let µ be a complex measure on a measurable space (Ω, F ). We call total variation of µ, the map |µ| : F → [0, +∞], defined by: +∞ |µ(En )| ∀E ∈ F , |µ|(E ) = sup n=1 where the ’sup’ is taken over all measurable partitions (En )n≥1 of E . Tutorial 11: Complex Measures 7 Exercise 4. Let µ be a complex measure on (Ω, F ). 1. Show that for all E ∈ F , |µ(E )| ≤ |µ|(E ). 2. Show that |µ|(∅) = 0. Exercise 5. Let µ be a complex measure on (Ω, F ). Let E ∈ F and (En )n≥1 be a measurable partition of E . 1. Show that there exists (tn )n≥1 in R, with tn < |µ|(En ) for all n. 2. Show that for all n ≥ 1, there exists a measurable partition p (En )p≥1 of En such that: +∞ p |µ(En )| tn < p=1 p 3. Show that (En )n,p≥1 is a measurable partition of E . 4. Show that for all N ≥ 1, we have N n=1 tn ≤ |µ|(E ). Tutorial 11: Complex Measures 8 5. Show that for all N ≥ 1, we have: N |µ|(En ) ≤ |µ|(E ) n=1 6. Suppose that (Ap )p≥1 is another arbitrary measurable partition of E . Show that for all p ≥ 1: +∞ |µ(Ap )| ≤ |µ(Ap ∩ En )| n=1 7. Show that for all n ≥ 1: +∞ |µ(Ap ∩ En )| ≤ |µ|(En ) p=1 8. Show that: +∞ +∞ |µ(Ap )| ≤ p=1 |µ|(En ) n=1 Tutorial 11: Complex Measures 9 9. Show that |µ| : F → [0, +∞] is a measure on (Ω, F ). Exercise 6. Let a, b ∈ R, a < b. Let F ∈ C 1 ([a, b]; R), and define: x ∀x ∈ [a, b] , H (x) = F (t)dt a 1. Show that H ∈ C 1 ([a, b]; R) and H = F . 2. Show that: b F (b) − F (a) = F (t)dt a 3. Show that: 1 2π +π/2 cos θdθ = −π/2 1 π 4. Let u ∈ R and τu : R → Rn be the translation τu (x) = x + u. Show that the Lebesgue measure dx on (Rn , B (Rn )) is invariant by translation τu , i.e. dx({τu ∈ B }) = dx(B ) for all B ∈ B (Rn ). n n Tutorial 11: Complex Measures 10 5. Show that for all f ∈ L1 (Rn , B (Rn ), dx), and u ∈ Rn : C f (x + u)dx = Rn f (x)dx Rn 6. Show that for all α ∈ R, we have: +π −π cos+ (α − θ)dθ = +π − α cos+ θdθ −π −α 7. Let α ∈ R and k ∈ Z such that k ≤ α/2π < k + 1. Show: −π − α ≤ −2kπ − π < π − α ≤ −2kπ + π 8. Show that: −2kπ −π −π −α cos+ θdθ = −2kπ +π π −α cos+ θdθ Tutorial 11: Complex Measures 11 9. Show that: +π − α −2kπ +π cos+ θdθ = −π −α +π cos+ θdθ = −2kπ −π cos+ θdθ −π 10. Show that for all α ∈ R: 1 2π +π −π cos+ (α − θ)dθ = 1 π Exercise 7. Let z1 , . . . , zN be N complex numbers. Let αk ∈ R be such that zk = |zk |eiαk , for all k = 1, . . . , N . For all θ ∈ [−π, +π ], we define S (θ) = {k = 1, . . . , N : cos(αk − θ) > 0}. 1. Show that for all θ ∈ [−π, +π ], we have: zk e−iθ ≥ zk = k ∈ S (θ ) k ∈ S (θ ) |zk | cos(αk − θ) k ∈ S (θ ) Tutorial 11: Complex Measures 12 N + 2. Define φ : [−π, +π ] → R by φ(θ) = k=1 |zk | cos (αk − θ ). Show the existence of θ0 ∈ [−π, +π ] such that: φ(θ0 ) = sup 3. Show that: 1 2π +π −π φ(θ) θ ∈[−π,+π ] 1 φ(θ)dθ = π N |zk | k=1 4. Conclude that: 1 π N |zk | ≤ k=1 zk k ∈ S (θ 0 ) Tutorial 11: Complex Measures 13 Exercise 8. Let µ ∈ M 1 (Ω, F ). Suppose that |µ|(E ) = +∞ for some E ∈ F . Define t = π (1 + |µ(E )|) ∈ R+ . 1. Show that there is a measurable partition (En )n≥1 of E , with: +∞ |µ(En )| t< n=1 2. Show the existence of N ≥ 1 such that: N |µ(En )| t< n=1 3. Show the existence of S ⊆ {1, . . . , N } such that: N |µ(En )| ≤ π n=1 µ(En ) n∈S 4. Show that |µ(A)| > t/π , where A = n∈S En . Tutorial 11: Complex Measures 14 5. Let B = E \ A. Show that |µ(B )| ≥ |µ(A)| − |µ(E )|. 6. Show that E = A B with |µ(A)| > 1 and |µ(B )| > 1. 7. Show that |µ|(A) = +∞ or |µ|(B ) = +∞. Exercise 9. Let µ ∈ M 1 (Ω, F ). Suppose that |µ|(Ω) = +∞. 1. Show the existence of A1 , B1 ∈ F , such that Ω = A1 |µ(A1 )| > 1 and |µ|(B1 ) = +∞. B1 , 2. Show the existence of a sequence (An )n≥1 of pairwise disjoint elements of F , such that |µ(An )| > 1 for all n ≥ 1. 3. Show that the series ∞ where A = +=1 An . n +∞ n=1 µ(An ) does not converge to µ(A) 4. Conclude that |µ|(Ω) < +∞. Tutorial 11: Complex Measures 15 Theorem 57 Let µ be a complex measure on a measurable space (Ω, F ). Then, its total variation |µ| is a finite measure on (Ω, F ). Exercise 10. Show that M 1 (Ω, F ) is a C-vector space, with: (λ + µ)(E ) = λ(E ) + µ(E ) (αλ)(E ) = α.λ(E ) where λ, µ ∈ M (Ω, F ), α ∈ C, and E ∈ F . 1 Definition 95 Let H be a K-vector space, where K = R or C. We call norm on H, any map N : H → R+ , with the following properties: (i) (ii) (iii) ∀x ∈ H , (N (x) = 0 ⇔ x = 0) ∀x ∈ H, ∀α ∈ K , N (αx) = |α|N (x) ∀x, y ∈ H , N (x + y ) ≤ N (x) + N (y ) Tutorial 11: Complex Measures 16 Exercise 11. 1. Explain why . 2. Show that · = p may not be a norm on Lp (Ω, F , µ). K ·, · is a norm, when ·, · is an inner-product. 3. Show that µ = |µ|(Ω) defines a norm on M 1 (Ω, F ). Exercise 12. Let µ ∈ M 1 (Ω, F ) be a signed measure. Show that: µ+ = µ− = 1 (|µ| + µ) 2 1 (|µ| − µ) 2 are finite measures such that: µ = µ+ − µ− , |µ| = µ+ + µ− Tutorial 11: Complex Measures 17 Exercise 13. Let µ ∈ M 1 (Ω, F ) and l : R2 → R be a linear map. 1. Show that l is continuous. 2. Show that l ◦ µ is a signed measure on (Ω, F ). 4 3. Show that all µ ∈ M 1 (Ω, F ) can be decomposed as: µ = µ1 − µ2 + i(µ3 − µ4 ) where µ1 , µ2 , µ3 , µ4 are finite measures. 4l ◦ µ refers strictly speaking to l(Re(µ), Im(µ)). Tutorial 12: Radon-Nikodym Theorem 1 12. Radon-Nikodym Theorem In the following, (Ω, F ) is an arbitrary measurable space. Definition 96 Let µ and ν be two (possibly complex) measures on (Ω, F ). We say that ν is absolutely continuous with respect to µ, and we write ν << µ, if and only if, for all E ∈ F : µ(E ) = 0 ⇒ ν (E ) = 0 Exercise 1. Let µ be a measure on (Ω, F ) and ν ∈ M 1 (Ω, F ). Show that ν << µ is equivalent to |ν | << µ. Exercise 2. Let µ be a measure on (Ω, F ) and ν ∈ M 1 (Ω, F ). Let > 0. Suppose there exists a sequence (En )n≥1 in F such that: ∀n ≥ 1 , µ(En ) ≤ 1 , |ν (En )| ≥ 2n Define: E = lim sup En = n≥1 Ek n≥1 k≥n Tutorial 12: Radon-Nikodym Theorem 2 1. Show that: µ(E ) = lim µ n→+∞ 2. Show that: Ek = 0 k ≥n |ν |(E ) = lim |ν | n→+∞ Ek ≥ k ≥n 3. Let λ be a measure on (Ω, F ). Can we conclude in general that: λ(E ) = lim λ n→+∞ 4. Prove the following: Ek k ≥n Tutorial 12: Radon-Nikodym Theorem 3 Theorem 58 Let µ be a measure on (Ω, F ) and ν be a complex measure on (Ω, F ). The following are equivalent: (i) (ii) (iii) ν << µ |ν | << µ ∀ > 0, ∃δ > 0, ∀E ∈ F , µ(E ) ≤ δ ⇒ |ν (E )| < Exercise 3. Let µ be a measure on (Ω, F ) and ν ∈ M 1 (Ω, F ) such that ν << µ. Let ν1 = Re(ν ) and ν2 = Im(ν ). 1. Show that ν1 << µ and ν2 << µ. +−+− 2. Show that ν1 , ν1 , ν2 , ν2 are absolutely continuous w.r. to µ. Exercise 4. Let µ be a finite measure on (Ω, F ) and f ∈ L1 (Ω, F , µ). C Let S be a closed subset of C. We assume that for all E ∈ F such that µ(E ) > 0, we have: 1 µ(E ) f dµ ∈ S E Tutorial 12: Radon-Nikodym Theorem 4 1. Show the existence of a sequence (Dn ) of closed discs in C with: +∞ Sc = Dn n=1 Let αn ∈ C, rn > 0 be such that Dn = {z ∈ C : |z − αn | ≤ rn }. 2. Suppose µ(En ) > 0 for some n ≥ 1, where En = {f ∈ Dn }. Show that: 1 1 f dµ − αn ≤ |f − αn |dµ ≤ rn µ(En ) En µ(En ) En 3. Show that for all n ≥ 1, µ({f ∈ Dn }) = 0. 4. Prove the following: Tutorial 12: Radon-Nikodym Theorem 5 Theorem 59 Let µ be a finite measure on (Ω, F ), f ∈ L1 (Ω, F , µ). C Let S be a closed subset of C such that for all E ∈ F with µ(E ) > 0, we have: 1 f dµ ∈ S µ(E ) E Then, f ∈ S µ-a.s. Exercise 5. Let µ be a σ -finite measure on (Ω, F ). Let (En )n≥1 be a sequence in F such that En ↑ Ω and µ(En ) < +∞ for all n ≥ 1. Define w : (Ω, F ) → (R, B (R)) as: +∞ w= 1 1 1E 2n 1 + µ(En ) n n=1 1. Show that for all ω ∈ Ω, 0 < w(ω ) ≤ 1. 2. Show that w ∈ L1 (Ω, F , µ). R Tutorial 12: Radon-Nikodym Theorem 6 Exercise 6. Let µ be a σ -finite measure on (Ω, F ) and ν be a finite measure on (Ω, F ), such that ν << µ. Let w ∈ L1 (Ω, F , µ) be such R that 0 < w ≤ 1. We define µ = wdµ, i.e. ¯ ∀E ∈ F , µ(E ) = ¯ wdµ E 1. Show that µ is a finite measure on (Ω, F ). ¯ 2. Show that φ = ν + µ is also a finite measure on (Ω, F ). ¯ 3. Show that for all f ∈ L1 (Ω, F , φ), we have f ∈ L1 (Ω, F , ν ), C C f w ∈ L1 (Ω, F , µ), and: C f dφ = Ω f dν + Ω f wdµ Ω 4. Show that for all f ∈ L2 (Ω, F , φ), we have: C |f |dν ≤ Ω |f |dφ ≤ Ω |f | dφ 2 Ω 1 2 1 (φ(Ω)) 2 Tutorial 12: Radon-Nikodym Theorem 7 5. Show that L2 (Ω, F , φ) ⊆ L1 (Ω, F , ν ), and for f ∈ L2 (Ω, F , φ): C C C f dν ≤ φ(Ω). f 2 Ω 6. Show the existence of g ∈ L2 (Ω, F , φ) such that: C ∀f ∈ L2 (Ω, F , φ) , C f dν = Ω f gdφ (1) Ω 7. Show that for all E ∈ F such that φ(E ) > 0, we have: 1 φ(E ) gdφ ∈ [0, 1] E 8. Show the existence of g ∈ L2 (Ω, F , φ) such that g (ω) ∈ [0, 1] C for all ω ∈ Ω, and (1) still holds. 9. Show that for all f ∈ L2 (Ω, F , φ), we have: C f (1 − g )dν = Ω f gwdµ Ω Tutorial 12: Radon-Nikodym Theorem 8 10. Show that for all n ≥ 1 and E ∈ F , f = (1 + g + . . . + g n )1E ∈ L2 (Ω, F , φ) C 11. Show that for all n ≥ 1 and E ∈ F , (1 − g n+1 )dν = E g (1 + g + . . . + g n )wdµ E 12. Define: +∞ gn h = gw n=0 Show that if A = {0 ≤ g < 1}, then for all E ∈ F : ν (E ∩ A) = hdµ E 13. Show that {h = +∞} = Ac and conclude that µ(Ac ) = 0. 14. Show that for all E ∈ F , we have ν (E ) = E hdµ. Tutorial 12: Radon-Nikodym Theorem 9 15. Show that if µ is σ -finite on (Ω, F ), and ν is a finite measure on (Ω, F ) such that ν << µ, there exists h ∈ L1 (Ω, F , µ), such R that h ≥ 0 and: ∀E ∈ F , ν (E ) = hdµ E 16. Prove the following: Theorem 60 (Radon-Nikodym:1) Let µ be a σ -finite measure on (Ω, F ). let ν be a complex measure on (Ω, F ) such that ν << µ. Then, there exists some h ∈ L1 (Ω, F , µ) such that: C ∀E ∈ F , ν (E ) = hdµ E If ν is a signed measure on (Ω, F ), we can assume h ∈ L1 (Ω, F , µ). R If ν is a finite measure on (Ω, F ), we can assume h ≥ 0. Tutorial 12: Radon-Nikodym Theorem 10 Exercise 7. Let f = u + iv ∈ L1 (Ω, F , µ), such that: C ∀E ∈ F , f dµ = 0 E where µ is a measure on (Ω, F ). 1. Show that: u+ dµ = Ω udµ {u≥0} 2. Show that f = 0 µ-a.s. 3. State and prove some uniqueness property in theorem (60). Exercise 8. Let µ and ν be two σ -finite measures on (Ω, F ) such that ν << µ. Let (En )n≥1 be a sequence in F such that En ↑ Ω and ν (En ) < +∞ for all n ≥ 1. We define: ∀n ≥ 1 , νn = ν En = ν (En ∩ ·) Tutorial 12: Radon-Nikodym Theorem 11 1. Show that there exists hn ∈ L1 (Ω, F , µ) with hn ≥ 0 and: R ∀E ∈ F , νn (E ) = hn dµ (2) E for all n ≥ 1. 2. Show that for all E ∈ F , hn dµ ≤ E hn+1 dµ E 3. Show that for all n, p ≥ 1, µ({hn − hn+1 > 1 }) = 0 p 4. Show that hn ≤ hn+1 µ-a.s. 5. Show the existence of a sequence (hn )n1 in L1 (Ω, F , µ) such R that 0 ≤ hn ≤ hn+1 for all n ≥ 1 and with (2) still holding. Tutorial 12: Radon-Nikodym Theorem 12 6. Let h = supn≥1 hn . Show that: ∀E ∈ F , ν (E ) = hdµ (3) E 7. Show that for all n ≥ 1, En hdµ < +∞. 8. Show that h < +∞ µ-a.s. 9. Show there exists h : (Ω, F ) → R+ measurable, while (3) holds. 10. Show that for all n ≥ 1, h ∈ L1 (Ω, F , µEn ). R Theorem 61 (Radon-Nikodym:2) Let µ and ν be two σ -finite measures on (Ω, F ) such that ν << µ. There exists a measurable map h : (Ω, F ) → (R+ , B (R+ )) such that: ∀E ∈ F , ν (E ) = hdµ E Tutorial 12: Radon-Nikodym Theorem 13 Exercise 9. Let h, h : (Ω, F ) → [0, +∞] be two non-negative and measurable maps. Let µ be a σ -finite measure on (Ω, F ). We assume: ∀E ∈ F , hdµ = E h dµ E Let (En )n≥1 be a sequence in F with En ↑ Ω and µ(En ) < +∞ for all n ≥ 1. We define Fn = En ∩ {h ≤ n} for all n ≥ 1. 1. Show that for all n and E ∈ F , E hdµFn = E h dµFn < +∞. 2. Show that for all n, p ≥ 1, µ(Fn ∩ {h > h + 1/p}) = 0. 3. Show that for all n ≥ 1, µ({Fn ∩ {h = h }) = 0. 4. Show that µ({h = h } ∩ {h < +∞}) = 0. 5. Show that h = h µ-a.s. 6. State and prove some uniqueness property in theorem (61). Tutorial 12: Radon-Nikodym Theorem 14 Exercise 10. Take Ω = {∗} and F = P (Ω) = {∅, {∗}}. Let µ be the measure on (Ω, F ) defined by µ(∅) = 0 and µ({∗}) = +∞. Let h, h : (Ω, F ) → [0, +∞] be defined by h(∗) = 1 = 2 = h (∗). Show that we have: hdµ = h dµ ∀E ∈ F , E E Explain why this does not contradict the previous exercise. Exercise 11. Let µ be a complex measure on (Ω, F ). 1. Show that µ << |µ|. 2. Show the existence of some h ∈ L1 (Ω, F , |µ|) such that: C hd|µ| ∀E ∈ F , µ(E ) = E 3. If µ is a signed measure, can we assume h ∈ L1 (Ω, F , |µ|)? R Tutorial 12: Radon-Nikodym Theorem 15 Exercise 12. Further to ex. (11), define Ar = {|h| < r} for all r > 0. 1. Show that for all measurable partition (En )n≥1 of Ar : +∞ |µ(En )| ≤ r|µ|(Ar ) n=1 2. Show that |µ|(Ar ) = 0 for all 0 < r < 1. 3. Show that |h| ≥ 1 |µ|-a.s. 4. Suppose that E ∈ F is such that |µ|(E ) > 0. Show that: 1 |µ|(E ) 5. Show that |h| ≤ 1 |µ|-a.s. 6. Prove the following: hd|µ| ≤ 1 E Tutorial 12: Radon-Nikodym Theorem 16 Theorem 62 For all complex measure µ on (Ω, F ), there exists h belonging to L1 (Ω, F , |µ|) such that |h| = 1 and: C hd|µ| ∀E ∈ F , µ(E ) = E If µ is a signed measure on (Ω, F ), we can assume h ∈ L1 (Ω, F , |µ|). R Exercise 13. Let A ∈ F , and (An )n≥1 be a sequence in F . 1. Show that if An ↑ A then 1An ↑ 1A . 2. Show that if An ↓ A then 1An ↓ 1A . 3. Show that if 1An → 1A , then for all µ ∈ M 1 (Ω, F ): µ(A) = lim µ(An ) n→+∞ Tutorial 12: Radon-Nikodym Theorem 17 Exercise 14. Let µ be a measure on (Ω, F ) and f ∈ L1 (Ω, F , µ). C 1. Show that ν = f dµ ∈ M 1 (Ω, F ). 2. Let h ∈ L1 (Ω, F , |ν |) be such that |h| = 1 and ν = C Show that for all E, F ∈ F : h1F d|ν | f 1F dµ = E hd|ν |. E 3. Show that if g : (Ω, F ) → (C, B (C)) is bounded and measurable: ∀E ∈ F , hgd|ν | f gdµ = E E 4. Show that: ¯ f hdµ ∀E ∈ F , |ν |(E ) = E 5. Show that for all E ∈ F , ¯ Re(f h)dµ ≥ 0 , E ¯ Im(f h)dµ = 0 E Tutorial 12: Radon-Nikodym Theorem 18 ¯ 6. Show that f h ∈ R+ µ-a.s. ¯ 7. Show that f h = |f | µ-a.s. 8. Prove the following: Theorem 63 Let µ be a measure on (Ω, F ) and f ∈ L1 (Ω, F , µ). C Then, ν = f dµ defined by: ∀E ∈ F , ν (E ) = f dµ E is a complex measure on (Ω, F ) with total variation: |f |dµ ∀E ∈ F , |ν |(E ) = E Tutorial 12: Radon-Nikodym Theorem 19 Exercise 15. Let µ ∈ M 1 (Ω, F ) be a signed measure. Suppose that h ∈ L1 (Ω, F , |µ|) is such that |h| = 1 and µ = hd|µ|. Define R A = {h = 1} and B = {h = −1}. 1. Show that for all E ∈ F , µ+ (E ) = 1 (1 E2 + h)d|µ|. 2. Show that for all E ∈ F , µ− (E ) = 1 (1 E2 − h)d|µ|. 3. Show that µ+ = µA = µ(A ∩ · ). 4. Show that µ− = −µB = −µ(B ∩ · ). Theorem 64 (Hahn Decomposition) Let µ be a signed measure on (Ω, F ). There exist A, B ∈ F , such that A ∩ B = ∅, Ω = A B and for all E ∈ F , µ+ (E ) = µ(A ∩ E ) and µ− (E ) = −µ(B ∩ E ). Tutorial 12: Radon-Nikodym Theorem 20 Definition 97 Let µ be a complex measure on (Ω, F ). We define: L1 (Ω, F , µ) = L1 (Ω, F , |µ|) C C and for all f ∈ L1 (Ω, F , µ), the lebesgue integral of f with respect C to µ, is defined as: f dµ = f hd|µ| where h ∈ L1 (Ω, F , |µ|) is such that |h| = 1 and µ = C hd|µ|. Exercise 16. Let µ be a complex measure on (Ω, F ). 1. Show that for all f : (Ω, F ) → (C, B (C)) measurable: f ∈ L1 (Ω, F , µ) ⇔ C 2. Show that for f ∈ L1 (Ω, F , µ), C |f |d|µ| < +∞ f dµ is unambiguously defined. 3. Show that for all E ∈ F , 1E ∈ L1 (Ω, F , µ) and C 1E dµ = µ(E ). Tutorial 12: Radon-Nikodym Theorem 21 4. Show that if µ is a finite measure, then |µ| = µ. 5. Show that if µ is a finite measure, definition (97) of integral and space L1 (Ω, F , µ) is consistent with that already known C for measures. 6. Show that L1 (Ω, F , µ) is a C-vector space and that: C (f + αg )dµ = f dµ + α for all f, g ∈ L1 (Ω, F , µ) and α ∈ C. C 7. Show that for all f ∈ L1 (Ω, F , µ), we have: C f dµ ≤ |f |d|µ| g dµ Tutorial 12: Radon-Nikodym Theorem 22 Exercise 17. Let µ, ν ∈ M 1 (Ω, F ), let α ∈ C. 1. Show that |αν | = |α|.|ν | 2. Show that |µ + ν | ≤ |µ| + |ν | 3. Show that L1 (Ω, F , µ) ∩ L1 (Ω, F , ν ) ⊆ L1 (Ω, F , µ + αν ) C C C 4. Show that for all E ∈ F : 1E d(µ + αν ) = 1E dµ + α 1E dν 5. Show that for all f ∈ L1 (Ω, F , µ) ∩ L1 (Ω, F , ν ): C C f d(µ + αν ) = f dµ + α f dν Tutorial 12: Radon-Nikodym Theorem 23 Exercise 18. Let µ = µ1 + iµ2 ∈ M 1 (Ω, F ). 1. Show that |µ1 | ≤ |µ| and |µ2 | ≤ |µ|. 2. Show that |µ| ≤ |µ1 | + |µ2 |. 3. Show that L1 (Ω, F , µ) = L1 (Ω, F , µ1 ) ∩ L1 (Ω, F , µ2 ). C C C 4. Show that: L1 (Ω, F , µ1 ) C L1 (Ω, F , µ2 ) C = L1 (Ω, F , µ+ ) ∩ L1 (Ω, F , µ− ) C C 1 1 = L1 (Ω, F , µ+ ) ∩ L1 (Ω, F , µ− ) C C 2 2 5. Show that for all f ∈ L1 (Ω, F , µ): C f dµ = f dµ+ − 1 f dµ− + i 1 f dµ+ − 2 f dµ− 2 Tutorial 12: Radon-Nikodym Theorem 24 Exercise 19. Let µ ∈ M 1 (Ω, F ). Let A ∈ F . Let h ∈ L1 (Ω, F , |µ|) C be such that |h| = 1 and µ = hd|µ|. Recall that µA = µ(A ∩ · ) and µ|A = µ|(F|A ) where F|A = {A ∩ E , E ∈ F } ⊆ F . 1. Show that we also have F|A = {E : E ∈ F , E ⊆ A}. 2. Show that µA ∈ M 1 (Ω, F ) and µ|A ∈ M 1 (A, F|A ). 3. Let E ∈ F and (En )n≥1 be a measurable partition of E . Show: +∞ |µA (En )| ≤ |µ|A (E ) n=1 4. Show that we have |µA | ≤ |µ|A . 5. Let E ∈ F and (En )n≥1 be a measurable partition of A ∩ E . Show that: +∞ |µ(En )| ≤ |µA |(A ∩ E ) n=1 Tutorial 12: Radon-Nikodym Theorem 25 6. Show that |µA |(Ac ) = 0. 7. Show that |µA | = |µ|A . 8. Let E ∈ F|A and (En )n≥1 be an F|A -measurable partition of E . Show that: +∞ |µ|A (En )| ≤ |µ||A (E ) n=1 9. Show that |µ|A | ≤ |µ||A . 10. Let E ∈ F|A ⊆ F and (En )n≥1 be a measurable partition of E . Show that (En )n≥1 is also an F|A -measurable partition of E , and conclude: +∞ |µ(En )| ≤ |µ|A |(E ) n=1 11. Show that |µ|A | = |µ||A . 12. Show that µA = hd|µA |. Tutorial 12: Radon-Nikodym Theorem 26 13. Show that h|A ∈ L1 (A, F|A , |µ|A |) and µ|A = C h|A d|µ|A |. 14. Show that for all f ∈ L1 (Ω, F , µ), we have: C f 1A ∈ L1 (Ω, F , µ) , f ∈ L1 (Ω, F , µA ) , f|A ∈ L1 (A, F|A , µ|A ) C C C and: f 1A dµ = f dµA = f|A dµ|A Definition 98 Let f ∈ L1 (Ω, F , µ) , where µ is a complex measure C on (Ω, F ). let A ∈ F . We call partial lebesgue integral of f with respect to µ over A, the integral denoted A f dµ, defined as: f dµ = A (f 1A )dµ = f dµA = (f|A )dµ|A where µA is the complex measure on (Ω, F ), µA = µ(A ∩ · ), f|A is the restriction of f to A and µ|A is the restriction of µ to F|A , the trace of F on A. Tutorial 12: Radon-Nikodym Theorem 27 Exercise 20. Prove the following: Theorem 65 Let f ∈ L1 (Ω, F , µ), where µ is a complex measure C on (Ω, F ). Then, ν = f dµ defined as: ∀E ∈ F , ν (E ) = f dµ E is a complex measure on (Ω, F ), with total variation: |f |d|µ| ∀E ∈ F , |ν |(E ) = E Moreover, for all measurable map g : (Ω, F ) → (C, B (C)), we have: g ∈ L1 (Ω, F , ν ) ⇔ gf ∈ L1 (Ω, F , µ) C C and when such condition is satisfied: g dν = g f dµ Tutorial 12: Radon-Nikodym Theorem 28 Exercise 21. Let (Ω1 , F1 ), . . . , (Ωn , Fn ) be n measurable spaces, where n ≥ 2. Let µ1 ∈ M 1 (Ω1 , F1 ), . . ., µn ∈ M 1 (Ωn , Fn ). For all i ∈ Nn , let hi belonging to L1 (Ωi , Fi , |µi |) be such that |hi | = 1 C and µi = hi d|µi |. For all E ∈ F1 ⊗ . . . ⊗ Fn , we define: h1 . . . hn d|µ1 | ⊗ . . . ⊗ |µn | µ(E ) = E 1. Show that µ ∈ M 1 (Ω1 × . . . × Ωn , F1 ⊗ . . . ⊗ Fn ) 2. Show that for all measurable rectangle A1 × . . . × An : µ(A1 × . . . × An ) = µ1 (A1 ) . . . µn (An ) 3. Prove the following: Tutorial 12: Radon-Nikodym Theorem 29 Theorem 66 Let µ1 , . . . , µn be n complex measures on measurable spaces (Ω1 , F1 ), . . . , (Ωn , Fn ) respectively, where n ≥ 2. There exists a unique complex measure µ1 ⊗ . . . ⊗ µn on (Ω1 × . . . × Ωn , F1 ⊗ . . . ⊗Fn ) such that for all measurable rectangle A1 × . . . × An , we have: µ1 ⊗ . . . ⊗ µn (A1 × . . . × An ) = µ1 (A1 ) . . . µn (An ) Exercise 22. Further to theorem (66), 1. Show that |µ1 ⊗ . . . ⊗ µn | = |µ1 | ⊗ . . . ⊗ |µn |. 2. Show that µ1 ⊗ . . . ⊗ µn = µ1 . . . µn . 3. Show that for all E ∈ F1 ⊗ . . . ⊗ Fn : µ1 ⊗ . . . ⊗ µn (E ) = h1 . . . hn d|µ1 ⊗ . . . ⊗ µn | E 4. Let f ∈ L1 (Ω1 × . . . × Ωn , F1 ⊗ . . . ⊗ Fn , µ1 ⊗ . . . ⊗ µn ). Show: C f dµ1 ⊗ . . . ⊗ µn = f h1 . . . hn d|µ1 | ⊗ . . . ⊗ |µn | Tutorial 12: Radon-Nikodym Theorem 30 5. let σ be a permutation of {1, . . . , n}. Show that: f dµ1 ⊗ . . . ⊗ µn = ... Ωσ(n) f dµσ(1) . . . dµσ(n) Ωσ(1) Tutorial 13: Regular Measure 1 13. Regular Measure In the following, K denotes R or C. Definition 99 Let (Ω, F ) be a measurable space. We say that a map s : Ω → C is a complex simple function on (Ω, F ), if and only if it is of the form: n αi 1Ai s= i=1 where n ≥ 1, αi ∈ C and Ai ∈ F for all i ∈ Nn . The set of all complex simple functions on (Ω, F ) is denoted SC (Ω, F ). The set of all R-valued complex simple functions in (Ω, F ) is denoted SR (Ω, F ). Recall that a simple function on (Ω, F ), as defined in (40), is just a non-negative element of SR (Ω, F ). Tutorial 13: Regular Measure 2 Exercise 1. Let (Ω, F , µ) be a measure space and p ∈ [1, +∞[. 1. Suppose s : Ω → C is of the form n αi 1Ai s= i=1 where n ≥ 1, αi ∈ C, Ai ∈ F and µ(Ai ) < +∞ for all i ∈ Nn . Show that s ∈ Lp (Ω, F , µ) ∩ SC (Ω, F ). C 2. Show that any s ∈ SC (Ω, F ) can be written as: n αi 1Ai s= i=1 where n ≥ 1, αi ∈ C \ {0}, Ai ∈ F and Ai ∩ Aj = ∅ for i = j . 3. Show that any s ∈ Lp (Ω, F , µ) ∩ SC (Ω, F ) is of the form: C n αi 1Ai s= i=1 Tutorial 13: Regular Measure 3 where n ≥ 1, αi ∈ C, Ai ∈ F and µ(Ai ) < +∞, for all i ∈ Nn . 4. Show that L∞ (Ω, F , µ) ∩ SC (Ω, F ) = SC (Ω, F ). C Exercise 2. Let (Ω, F , µ) be a measure space and p ∈ [1, +∞[. Let f be a non-negative element of Lp (Ω, F , µ). R 1. Show the existence of a sequence (sn )n≥1 of non-negative functions in Lp (Ω, F , µ) ∩ SR (Ω, F ) such that sn ↑ f . R 2. Show that: lim n→+∞ |sn − f |p dµ = 0 3. Show that there exists s ∈ Lp (Ω, F , µ) ∩ SR (Ω, F ) such that R f − s p ≤ , for all > 0. 4. Show that Lp (Ω, F , µ) ∩ SK (Ω, F ) is dense in Lp (Ω, F , µ). K K Tutorial 13: Regular Measure 4 Exercise 3. Let (Ω, F , µ) be a measure space. Let f be a nonnegative element of L∞ (Ω, F , µ). For all n ≥ 1, we define: R n2n −1 sn = k=0 k n n + n1{n≤f } 1 2n {k/2 ≤f <(k+1)/2 } 1. Show that for all n ≥ 1, sn is a simple function. 2. Show there exists n0 ≥ 1 and N ∈ F with µ(N ) = 0, such that: ∀ω ∈ N c , 0 ≤ f (ω ) < n0 3. Show that for all n ≥ n0 and ω ∈ N c , we have: 1 0 ≤ f (ω ) − sn (ω ) < n 2 4. Conclude that: lim n→+∞ 5. Show the following: f − sn ∞ =0 Tutorial 13: Regular Measure 5 Theorem 67 Let (Ω, F , µ) be a measure space and p ∈ [1, +∞]. Then, Lp (Ω, F , µ) ∩ SK (Ω, F ) is dense in Lp (Ω, F , µ). K K Exercise 4. Let (Ω, T ) be a metrizable topological space, and µ be a finite measure on (Ω, B (Ω)). We define Σ as the set of all B ∈ B (Ω) such that for all > 0, there exist F closed and G open in Ω, with: F ⊆ B ⊆ G , µ(G \ F ) ≤ Given a metric d on (Ω, T ) inducing the topology T , we define: d(x, A) = inf {d(x, y ) : y ∈ A} for all A ⊆ Ω and x ∈ Ω. ¯ 1. Show that x → d(x, A) from Ω to R is continuous for all A ⊆ Ω. 2. Show that if F is closed in Ω, x ∈ F is equivalent to d(x, F ) = 0. Tutorial 13: Regular Measure 6 Exercise 5. Further to exercise (4), we assume that F is a closed subset of Ω. For all n ≥ 1, we define: Gn = {x ∈ Ω : d(x, F ) < 1 } n 1. Show that Gn is open for all n ≥ 1. 2. Show that Gn ↓ F . 3. Show that F ∈ Σ. 4. Was it important to assume that µ is finite? 5. Show that Ω ∈ Σ. 6. Show that if B ∈ Σ, then B c ∈ Σ. Tutorial 13: Regular Measure 7 Exercise 6. Further to exercise (5), let (Bn )n≥1 be a sequence in Σ. ∞ Define B = ∪+=1 Bn and let > 0. n 1. Show that for all n, there is Fn closed and Gn open in Ω, with: Fn ⊆ Bn ⊆ Gn , µ(Gn \ Fn ) ≤ 2n 2. Show the existence of some N ≥ 1 such that: +∞ N Fn µ \ n=1 ≤ Fn n=1 ∞ 3. Define G = ∪+=1 Gn and F = ∪N=1 Fn . Show that F is closed, n n G is open and F ⊆ B ⊆ G. 4. Show that: +∞ G\F ⊆ G\ +∞ Fn n=1 Fn n=1 \F Tutorial 13: Regular Measure 5. Show that: 8 +∞ G\ +∞ Fn n=1 Gn \ Fn ⊆ n=1 6. Show that µ(G \ F ) ≤ 2 . 7. Show that Σ is a σ -algebra on Ω, and conclude that Σ = B (Ω). Theorem 68 Let (Ω, T ) be a metrizable topological space, and µ be a finite measure on (Ω, B (Ω)). Then, for all B ∈ B (Ω) and > 0, there exist F closed and G open in Ω such that: F ⊆ B ⊆ G , µ(G \ F ) ≤ b Definition 100 Let (Ω, T ) be a topological space. We denote CK (Ω) the K-vector space of all continuous, bounded maps φ : Ω → K, where K = R or K = C. Tutorial 13: Regular Measure 9 Exercise 7. Let (Ω, T ) be a metrizable topological space with some metric d. Let µ be a finite measure on (Ω, B (Ω)) and F be a closed subset of Ω. For all n ≥ 1, we define φn : Ω → R by: ∀x ∈ Ω , φn (x) = 1 − 1 ∧ (nd(x, F )) b 1. Show that for all p ∈ [1, +∞], we have CK (Ω) ⊆ Lp (Ω, B (Ω), µ). K b 2. Show that for all n ≥ 1, φn ∈ CR (Ω). 3. Show that φn → 1F . 4. Show that for all p ∈ [1, +∞[, we have: lim n→+∞ |φn − 1F |p dµ = 0 b 5. Show that for all p ∈ [1, +∞[ and > 0, there exists φ ∈ CR (Ω) such that φ − 1F p ≤ . Tutorial 13: Regular Measure 10 b 6. Let ν ∈ M 1 (Ω, B (Ω)). Show that CC (Ω) ⊆ L1 (Ω, B (Ω), ν ) and: C ν (F ) = lim n→+∞ φn dν 7. Prove the following: Theorem 69 Let (Ω, T ) be a metrizable topological space and µ, ν be two complex measures on (Ω, B (Ω)) such that: b ∀φ ∈ CR (Ω) , φdµ = φdν Then µ = ν . Exercise 8. Let (Ω, T ) be a metrizable topological space and µ be a finite measure on (Ω, B (Ω)). Let s ∈ SC (Ω, B (Ω)) be a complex simple function: n αi 1Ai s= i=1 Tutorial 13: Regular Measure 11 where n ≥ 1, αi ∈ C, Ai ∈ B (Ω) for all i ∈ Nn . Let p ∈ [1, +∞[. 1. Show that given > 0, for all i ∈ Nn there is a closed subset Fi of Ω such that Fi ⊆ Ai and µ(Ai \ Fi ) ≤ . Let: n s= αi 1Fi i=1 2. Show that: n s−s p ≤ |αi | 1 p i=1 b 3. Conclude that given > 0, there exists φ ∈ CC (Ω) such that: φ−s 4. Prove the following: p ≤ Tutorial 13: Regular Measure 12 Theorem 70 Let (Ω, T ) be a metrizable topological space and µ be b a finite measure on (Ω, B (Ω)). Then, for all p ∈ [1, +∞[, CK (Ω) is p dense in LK (Ω, B (Ω), µ). Definition 101 A topological space (Ω, T ) is said to be σ -compact if and only if, there exists a sequence (Kn )n≥1 of compact subsets of Ω such that Kn ↑ Ω. Exercise 9. Let (Ω, T ) be a metrizable and σ -compact topological space, with metric d. Let Ω be open in Ω. For all n ≥ 1, we define: Fn = {x ∈ Ω : d(x, (Ω )c ) ≥ 1/n} Let (Kn )n≥1 be a sequence of compact subsets of Ω such that Kn ↑ Ω. 1. Show that for all n ≥ 1, Fn is closed in Ω. 2. Show that Fn ↑ Ω . Tutorial 13: Regular Measure 13 3. Show that Fn ∩ Kn ↑ Ω . 4. Show that Fn ∩ Kn is closed in Kn for all n ≥ 1. 5. Show that Fn ∩ Kn is a compact subset of Ω for all n ≥ 1 6. Prove the following: Theorem 71 Let (Ω, T ) be a metrizable and σ -compact topological space. Then, for all Ω open subset of Ω, the induced topological space (Ω , T|Ω ) is itself metrizable and σ -compact. Definition 102 Let (Ω, T ) be a topological space and µ be a measure on (Ω, B (Ω)). We say that µ is locally finite, if and only if, every x ∈ Ω has an open neighborhood of finite µ-measure, i.e. ∀x ∈ Ω , ∃U ∈ T , x ∈ U , µ(U ) < +∞ Tutorial 13: Regular Measure 14 Definition 103 Let µ be a measure on a topological space (Ω, T ). We say that µ is inner-regular, if and only if, for all B ∈ B (Ω): µ(B ) = sup{µ(K ) : K ⊆ B , K compact} We say that µ is outer-regular, if and only if, for all B ∈ B (Ω): µ(B ) = inf {µ(G) : B ⊆ G , G open} We say that µ is regular if it is both inner and outer-regular. Exercise 10. Let (Ω, T ) be a topological space and µ be a locally finite measure on (Ω, B (Ω)). Let K be a compact subset of Ω. 1. Show the existence of open sets V1 , . . . , Vn with µ(Vi ) < +∞ for all i ∈ Nn and K ⊆ V1 ∪ . . . ∪ Vn , where n ≥ 1. 2. Conclude that µ(K ) < +∞. Tutorial 13: Regular Measure 15 Exercise 11. Let (Ω, T ) be a metrizable and σ -compact topological space. Let µ be a finite measure on (Ω, B (Ω)). Let (Kn )n≥1 be a sequence of compact subsets of Ω such that Kn ↑ Ω. Let B ∈ B (Ω). We define α = sup{µ(K ) : K ⊆ B , K compact}. 1. Show that given > 0, there exists F closed in Ω such that F ⊆ B and µ(B \ F ) ≤ . 2. Show that F \ (Kn ∩ F ) ↓ ∅. 3. Show that Kn ∩ F is closed in Kn . 4. Show that Kn ∩ F is compact. 5. Conclude that given > 0, there exists K compact subset of Ω such that K ⊆ F and µ(F \ K ) ≤ . 6. Show that µ(B ) ≤ µ(K ) + 2 . 7. Show that µ(B ) ≤ α and conclude that µ is inner-regular. Tutorial 13: Regular Measure 16 Exercise 12. Let (Ω, T ) be a metrizable and σ -compact topological space. Let µ be a locally finite measure on (Ω, B (Ω)). Let (Kn )n≥1 be a sequence of compact subsets of Ω such that Kn ↑ Ω. Let B ∈ B (Ω), and α ∈ R be such that α < µ(B ). 1. Show that µ(Kn ∩ B ) ↑ µ(B ). 2. Show the existence of K ⊆ Ω compact, with α < µ(K ∩ B ). 3. Let µK = µ(K ∩ · ). Show that µK is a finite measure, and conclude that µK (B ) = sup{µK (K ∗ ) : K ∗ ⊆ B , K ∗ compact}. 4. Show the existence of a compact subset K ∗ of Ω, such that K ∗ ⊆ B and α < µ(K ∩ K ∗ ). 5. Show that K ∗ is closed in Ω. 6. Show that K ∩ K ∗ is closed in K . 7. Show that K ∩ K ∗ is compact. Tutorial 13: Regular Measure 17 8. Show that α < sup{µ(K ) : K ⊆ B , K compact}. 9. Show that µ(B ) ≤ sup{µ(K ) : K ⊆ B , K compact}. 10. Conclude that µ is inner-regular. Exercise 13. Let (Ω, T ) be a metrizable topological space. 1. Show that (Ω, T ) is separable if and only if it has a countable base. 2. Show that if (Ω, T ) is compact, for all n ≥ 1, Ω can be covered by a finite number of open balls with radius 1/n. 3. Show that if (Ω, T ) is compact, then it is separable. Tutorial 13: Regular Measure 18 Exercise 14. Let (Ω, T ) be a metrizable and σ -compact topological space with metric d. Let (Kn )n≥1 be a sequence of compact subsets of Ω such that Kn ↑ Ω. 1. For all n ≥ 1, give a metric on Kn inducing the topology T|Kn . 2. Show that (Kn , T|Kn ) is separable. Let (xp )p≥1 be a countable n dense family of (Kn , T|Kn ). 3. Show that (xp )n,p≥1 is a countable dense family of (Ω, T ), and n conclude with the following: Theorem 72 Let (Ω, T ) be a metrizable and σ -compact topological space. Then, (Ω, T ) is separable and has a countable base. Exercise 15. Let (Ω, T ) be a metrizable and σ -compact topological space. Let µ be a locally finite measure on (Ω, B (Ω)). Let H be a countable base of (Ω, T ). We define H = {V ∈ H : µ(V ) < +∞}. Tutorial 13: Regular Measure 19 1. Show that for all U open in Ω and x ∈ U , there is Ux open in Ω such that x ∈ Ux ⊆ U and µ(Ux ) < +∞. 2. Show the existence of Vx ∈ H such that x ∈ Vx ⊆ Ux . 3. Conclude that H is a countable base of (Ω, T ). 4. Show the existence of a sequence (Vn )n≥1 of open sets in Ω with: +∞ Vn , µ(Vn ) < +∞ , ∀n ≥ 1 Ω= n=1 Exercise 16. Let (Ω, T ) be a metrizable and σ -compact topological space. Let µ be a locally finite measure on (Ω, B (Ω)). Let (Vn )n≥1 a sequence of open subsets of Ω such that: +∞ Vn , µ(Vn ) < +∞ , ∀n ≥ 1 Ω= n=1 Let B ∈ B (Ω) and α = inf {µ(G) : B ⊆ G , G open}. Tutorial 13: Regular Measure 20 1. Given > 0, show that there exists Gn open in Ω such that B ⊆ Gn and µVn (Gn \ B ) ≤ /2n , where µVn = µ(Vn ∩ · ). ∞ 2. Let G = ∪+=1 (Vn ∩ Gn ). Show that G is open in Ω, and B ⊆ G. n ∞ 3. Show that G \ B = ∪+=1 Vn ∩ (Gn \ B ). n 4. Show that µ(G) ≤ µ(B ) + . 5. Show that α ≤ µ(B ). 6. Conclude that is µ outer-regular. 7. Show the following: Theorem 73 Let µ be a locally finite measure on a metrizable and σ -compact topological space (Ω, T ). Then, µ is regular, i.e.: µ(B ) = sup{µ(K ) : K ⊆ B , K compact} = inf {µ(G) : B ⊆ G , G open} for all B ∈ B (Ω). Tutorial 13: Regular Measure 21 Exercise 17. Show the following: Theorem 74 Let Ω be an open subset of Rn , where n ≥ 1. Any locally finite measure on (Ω, B (Ω)) is regular. Definition 104 We call strongly σ -compact topological space, a topological space (Ω, T ), for which there exists a sequence (Vn )n≥1 of open sets with compact closure, such that Vn ↑ Ω. Definition 105 We call locally compact topological space, a topological space (Ω, T ), for which every x ∈ Ω has an open neighborhood with compact closure, i.e. such that: ¯ ∀x ∈ Ω , ∃U ∈ T : x ∈ U , U is compact Tutorial 13: Regular Measure 22 Exercise 18. Let (Ω, T ) be a σ -compact and locally compact topological space. Let (Kn )n≥1 be a sequence of compact subsets of Ω such that Kn ↑ Ω. 1. Show that for all n ≥ 1, there are open sets V1n , . . . , Vpn , pn ≥ 1, n ¯ ¯ such that Kn ⊆ V1n ∪ . . . ∪ Vpn and V1n , . . . , Vpn are compact n n subsets of Ω. 2. Define Wn = V1n ∪ . . . ∪ Vpn and Vn = ∪n=1 Wk , for n ≥ 1. Show k n that (Vn )n≥1 is a sequence of open sets in Ω with Vn ↑ Ω. ¯ ¯ ¯ 3. Show that Wn = V1n ∪ . . . ∪ Vpn for all n ≥ 1. n ¯ 4. Show that Wn is compact for all n ≥ 1. ¯ 5. Show that Vn is compact for all n ≥ 1 6. Conclude with the following: Tutorial 13: Regular Measure 23 Theorem 75 A topological space (Ω, T ) is strongly σ -compact, if and only if it is σ -compact and locally compact. Exercise 19. Let (Ω, T ) be a topological space and Ω be an open ¯ subset of Ω. Let A ⊆ Ω . We denote AΩ the closure of A in the ¯ induced topological space (Ω , T|Ω ), and A its closure in Ω. ¯ 1. Show that A ⊆ Ω ∩ A. ¯ 2. Show that Ω ∩ A is closed in Ω . ¯ ¯ 3. Show that AΩ ⊆ Ω ∩ A. ¯ 4. Let x ∈ Ω ∩ A. Show that if x ∈ U ∈ T|Ω , then A ∩ U = ∅. ¯ ¯ 5. Show that AΩ = Ω ∩ A. Tutorial 13: Regular Measure 24 Exercise 20. Let (Ω, d) be a metric space. 1. Show that for all x ∈ Ω and > 0, we have: B (x, ) ⊆ {y ∈ Ω : d(x, y ) ≤ } 2. Take Ω = [0, 1/2[∪{1}. Show that B (0, 1) = [0, 1/2[. 3. Show that [0, 1/2[ is closed in Ω. 4. Show that B (0, 1) = [0, 1/2[. 5. Conclude that B (0, 1) = {y ∈ Ω : |y | ≤ 1} = Ω. Exercise 21. Let (Ω, d) be a locally compact metric space. Let Ω be an open subset of Ω. Let x ∈ Ω . 1. Show there exists U open with compact closure, such that x ∈ U . 2. Show the existence of > 0 such that B (x, ) ⊆ U ∩ Ω . Tutorial 13: Regular Measure 25 ¯ 3. Show that B (x, /2) ⊆ U . ¯ 4. Show that B (x, /2) is closed in U . 5. Show that B (x, /2) is a compact subset of Ω. 6. Show that B (x, /2) ⊆ Ω . 7. Let U = B (x, /2) ∩ Ω = B (x, /2). Show x ∈ U ∈ T|Ω , and: ¯ U Ω = B (x, /2) 8. Show that the induced topological space Ω is locally compact. 9. Prove the following: Theorem 76 Let (Ω, T ) be a metrizable and strongly σ -compact topological space. Then, for all Ω open subset of Ω, the induced topological space (Ω , T|Ω ) is itself metrizable and strongly σ -compact. Tutorial 13: Regular Measure 26 Definition 106 Let (Ω, T ) be a topological space, and φ : Ω → C be a map. We call support of φ, the closure of the set {φ = 0}, i.e.: supp(φ) = {ω ∈ Ω : φ(ω ) = 0} c Definition 107 Let (Ω, T ) be a topological space. We denote CK (Ω) the K-vector space of all continuous map with compact support φ : Ω → K, where K = R or K = C. Exercise 22. Let (Ω, T ) be a topological space. c 1. Show that 0 ∈ CK (Ω). c 2. Show that CK (Ω) is indeed a K-vector space. c b 3. Show that CK (Ω) ⊆ CK (Ω). Exercise 23. let (Ω, d) be a locally compact metric space. let K be a compact subset of Ω, and G be open in Ω, with K ⊆ G. Tutorial 13: Regular Measure 27 1. Show the existence of open sets V1 , . . . , Vn such that: K ⊆ V1 ∪ . . . ∪ Vn ¯ ¯ and V1 , . . . , Vn are compact subsets of Ω. 2. Show that V = (V1 ∪ . . . ∪ Vn ) ∩ G is open in Ω, and K ⊆ V ⊆ G. ¯ ¯ ¯ 3. Show that V ⊆ V1 ∪ . . . ∪ Vn . ¯ 4. Show that V is compact. 5. We assume K = ∅ and V = Ω, and we define φ : Ω → R by: ∀x ∈ Ω , φ(x) = d(x, V c ) d(x, V c ) + d(x, K ) 6. Show that φ is well-defined and continuous. 7. Show that {φ = 0} = V . c 8. Show that φ ∈ CR (Ω). Tutorial 13: Regular Measure 28 9. Show that 1K ≤ φ ≤ 1G . c 10. Show that if K = ∅, there is φ ∈ CR (Ω) with 1K ≤ φ ≤ 1G . 11. Show that if V = Ω then Ω is compact. c 12. Show that if V = Ω, there φ ∈ CR (Ω) with 1K ≤ φ ≤ 1G . Theorem 77 Let (Ω, T ) be a metrizable and locally compact topological space. Let K be a compact subset of Ω, and G be an open subset c of Ω such that K ⊆ G. Then, there exists φ ∈ CR (Ω), real-valued continuous map with compact support, such that: 1K ≤ φ ≤ 1G Exercise 24. Let (Ω, T ) be a metrizable and strongly σ -compact topological space. Let µ be a locally finite measure on (Ω, B (Ω)). Let B ∈ B (Ω) be such that µ(B ) < +∞. Let p ∈ [1, +∞[. c 1. Show that CK (Ω) ⊆ Lp (Ω, B (Ω), µ). K Tutorial 13: Regular Measure 29 2. Let > 0. Show the existence of K compact and G open, with: K ⊆ B ⊆ G , µ(G \ K ) ≤ 3. Where did you use the fact that µ(B ) < +∞? c 4. Show the existence of φ ∈ CR (Ω) with 1K ≤ φ ≤ 1G . 5. Show that: |φ − 1B |p dµ ≤ µ(G \ K ) 6. Conclude that for all c > 0, there exists φ ∈ CR (Ω) such that: φ − 1B p ≤ 7. Let s ∈ SC (Ω, B (Ω)) ∩ Lp (Ω, B (Ω), µ). Show that for all C c there exists φ ∈ CC (Ω) such that φ − s p ≤ . 8. Prove the following: > 0, Tutorial 13: Regular Measure 30 Theorem 78 Let (Ω, T ) be a metrizable and strongly σ -compact topological space1 . Let µ be a locally finite measure on (Ω, B (Ω)). c Then, for all p ∈ [1, +∞[, the space CK (Ω) of K-valued, continuous maps with compact support, is dense in Lp (Ω, B (Ω), µ). K Exercise 25. Prove the following: Theorem 79 Let Ω be an open subset of Rn , where n ≥ 1. Then, c for any locally finite measure µ on (Ω, B (Ω)) and p ∈ [1, +∞[, CK (Ω) p is dense in LK (Ω, B (Ω), µ). 1 i.e. a metrizable topological space for which there exists a sequence (Vn )n≥1 of open sets with compact closure, such that Vn ↑ Ω. Tutorial 14: Maps of Finite Variation 1 14. Maps of Finite Variation Definition 108 We call total variation of a map b : R+ → C the map |b| : R+ → [0, +∞] defined as: n ∀t ∈ R+ , |b|(t) = |b(0)| + sup |b(ti ) − b(ti−1 )| i=1 where the ’sup’ is taken over all finite t0 ≤ . . . ≤ tn in [0, t], n ≥ 1. We say that b is of finite variation, if and only if: ∀t ∈ R+ , |b|(t) < +∞ We say that b is of bounded variation, if and only if: sup |b|(t) < +∞ t∈R+ Warning: The notation |b| can be misleading: it can refer to the map t → |b(t)|(modulus), or to the map t → |b|(t) (total variation). Tutorial 14: Maps of Finite Variation 2 Exercise 1. Let a : R+ → R+ be non-decreasing with a(0) ≥ 0. 1. Show that |a| = a and a is of finite variation. ¯ 2. Show that the limit a(∞) = limt→+∞ a(t) exists in R. 3. Show that a is of bounded variation if and only if a(∞) < +∞. Exercise 2. Let b = b1 + ib2 : R+ → C be a map. 1. Show that |b1 | ≤ |b| and |b2 | ≤ |b|. 2. Show that |b| ≤ |b1 | + |b2 |. 3. Show that b is of finite variation if and only if b1 , b2 are. 4. Show that b is of bounded variation if and only if b1 , b2 are. 5. Show that |b|(0) = |b(0)|. Tutorial 14: Maps of Finite Variation 3 Exercise 3. Let b : R+ → R be continuous and differentiable, such that b is bounded on each compact interval. Show that b is of finite variation. Exercise 4. Show that if b : R+ → C is of class C 1 , i.e. continuous and differentiable with continuous derivative, then b is of finite variation. Exercise 5. Let f : (R+ , B (R+ )) → (C, B (C)) be a measurable map, t with 0 |f (s)|ds < +∞ for all t ∈ R+ . Let b : R+ → C defined by: ∀t ∈ R+ , b(t) = f 1[0,t] ds R+ 1. Show that b is of finite variation and: t ∀t ∈ R+ , |b|(t) ≤ |f (s)|ds 0 2. Show that f ∈ L1 (R+ , B (R+ ), ds) ⇒ b is of bounded variation. C Tutorial 14: Maps of Finite Variation 4 Exercise 6. Show that if b, b : R+ → C are maps of finite variation, and α ∈ C, then b + αb is also a map of finite variation. Prove the same result when the word ’finite’ is replaced by ’bounded’. Exercise 7. Let b : R+ → C be a map. For all t ∈ R+ , let S (t) be the set of all finite subsets A of [0, t], with cardA ≥ 2. For all A ∈ S (t), we define: n |b(ti ) − b(ti−1 )| S (A) = i=1 where it is understood that t0 , . . . , tn are such that: t0 < t1 < . . . < tn and A = {t0 , . . . , tn } ⊆ [0, t] 1. Show that for all t ∈ R+ , if s0 ≤ . . . ≤ sp (p ≥ 1) is a finite sequence in [0, t], then if: p |b(sj ) − b(sj −1 )| S= j =1 Tutorial 14: Maps of Finite Variation 5 either S = 0 or S = S (A) for some A ∈ S (t). 2. Conclude that: ∀t ∈ R+ , |b|(t) = |b(0)| + sup{S (A) : A ∈ S (t)} 3. Let A ∈ S (t) and s ∈ [0, t]. Show that S (A) ≤ S (A ∪ {s}). 4. Let A, B ∈ S (t). Show that: A ⊆ B ⇒ S (A) ≤ S (B ) 5. Show that if t0 ≤ . . . ≤ tn , n ≥ 1, and s0 ≤ . . . ≤ sp , p ≥ 1, are finite sequence in [0, t] such that: {t0 , . . . , tn } ⊆ {s0 , . . . , sp } then: p n |b(ti ) − b(ti−1 )| ≤ i=1 |b(sj ) − b(sj −1 )| j =1 Tutorial 14: Maps of Finite Variation 6 Exercise 8. Let b : R+ → C be of finite variation. Let s, t ∈ R+ , with s ≤ t. We define: n |b(ti ) − b(ti−1 )| δ = sup i=1 where the ’sup’ is taken over all finite t0 ≤ . . . ≤ tn , n ≥ 1, in [s, t]. 1. let s0 ≤ . . . ≤ sp and t0 ≤ . . . ≤ tn be finite sequences in [0, s] and [s, t] respectively, where n, p ≥ 1. Show that: p n |b(sj ) − b(sj −1 )| + |b(t0 ) − b(sp )| + j =1 |b(ti ) − b(ti−1 )| i=1 is less or equal than |b|(t) − |b(0)|. 2. Show that δ ≤ |b|(t) − |b|(s). 3. Let t0 ≤ . . . ≤ tn be a finite sequence in [0, t], where n ≥ 1, and Tutorial 14: Maps of Finite Variation 7 suppose that s = tj for some 0 < j < n. Show that: n |b(ti ) − b(ti−1 )| ≤ |b|(s) − |b(0)| + δ (1) i=1 4. Show that inequality (1) holds, for all t0 ≤ . . . ≤ tn in [0, t]. 5. Prove the following: Theorem 80 Let b : R+ → C be a map of finite variation. Then, for all s, t ∈ R+ , s ≤ t, we have: n |b(ti ) − b(ti−1 )| |b|(t) − |b|(s) = sup i=1 where the ’sup’ is taken over all finite t0 ≤ . . . ≤ tn , n ≥ 1, in [s, t]. Exercise 9. Let b : R+ → C be a map of finite variation. Show that |b| is non-decreasing with |b|(0) ≥ 0, and ||b|| = |b|. Tutorial 14: Maps of Finite Variation 8 Definition 109 Let b : R+ → R be a map of finite variation. Let: |b|+ = |b|− = 1 (|b| + b) 2 1 (|b| − b) 2 |b|+ , |b|− are respectively the positive, negative variation of b. Exercise 10. Let b : R+ → R be a map of finite variation. 1. Show that |b| = |b|+ + |b|− and b = |b|+ − |b|− . 2. Show that |b|+ (0) = b+ (0) ≥ 0 and |b|− (0) = b− (0) ≥ 0. 3. Show that for all s, t ∈ R+ , s ≤ t, we have: |b(t) − b(s)| ≤ |b|(t) − |b|(s) 4. Show that |b|+ and |b|− are non-decreasing. Tutorial 14: Maps of Finite Variation 9 Exercise 11. Let b : R+ → C be of finite variation. Show the existence of b1 , b2 , b3 , b4 : R+ → R+ , non-decreasing with bi (0) ≥ 0, such that b = b1 − b2 + i(b3 − b4 ). Show conversely that if b : R+ → C is a map with such decomposition, then it is of finite variation. Exercise 12. Let b : R+ → C be a right-continuous map of finite variation, and x0 ∈ R+ . 1. Show that |b|(x0 +) = limt↓↓x0 |b|(t) = inf x0 <t |b|(t) ∈ R. 2. Show that |b|(x0 ) ≤ |b|(x0 +). 3. Given > 0, show the existence of y0 ∈ R+ , x0 < y0 , such that: u ∈]x0 , y0 ] ⇒ |b(u) − b(x0 )| ≤ /2 u ∈]x0 , y0 ] ⇒ |b|(y0 ) − |b|(u) ≤ /2 Tutorial 14: Maps of Finite Variation 10 Exercise 13. Further to exercise (12), let t0 ≤ . . . ≤ tn , n ≥ 1, be a finite sequence in [0, y0 ], such that x0 = tj for some 0 < j < n − 1. We choose j to be the maximum index satisfying this condition, so that x0 < tj +1 ≤ y0 . 1. Show that j i=1 |b(ti ) − b(ti−1 )| ≤ |b|(x0 ) − |b(0)|. 2. Show that |b(tj +1 ) − b(tj )| ≤ /2. 3. Show that n i=j +2 |b(ti ) − b(ti−1 )| ≤ |b|(y0 ) − |b|(tj +1 ) ≤ /2. 4. Show that for all finite sequence t0 ≤ . . . ≤ tn , n ≥ 1, in [0, y0 ]: n |b(ti ) − b(ti−1 )| ≤ |b|(x0 ) − |b(0)| + i=1 5. Show that |b|(y0 ) ≤ |b|(x0 ) + . 6. Show that |b|(x0 +) ≤ |b|(x0 ) and that |b| is right-continuous. Tutorial 14: Maps of Finite Variation 11 Exercise 14. Let b : R+ → C be a left-continuous map of finite variation, and let x0 ∈ R+ \ {0}. 1. Show that |b|(x0 −) = limt↑↑x0 |b|(t) = supt<x0 |b|(t) ∈ R. 2. Show that |b|(x0 −) ≤ |b|(x0 ). 3. Given > 0, show the existence of y0 ∈]0, x0 [, such that: u ∈ [y0 , x0 [ ⇒ |b(x0 ) − b(u)| ≤ /2 u ∈ [y0 , x0 [ ⇒ |b|(u) − |b|(y0 ) ≤ /2 Exercise 15. Further to exercise (14), let t0 ≤ . . . ≤ tn , n ≥ 1, be a finite sequence in [0, x0 ], such that y0 = tj for some 0 < j < n − 1, and x0 = tn . We denote k = max{i : j ≤ i , ti < x0 }. 1. Show that j ≤ k ≤ n − 1 and tk ∈ [y0 , x0 [. 2. Show that j i=1 |b(ti ) − b(ti−1 )| ≤ |b|(y0 ) − |b(0)|. Tutorial 14: Maps of Finite Variation 12 k 3. Show that i=j +1 |b(ti ) − b(ti−1 )| ≤ |b|(tk ) − |b|(y0 ) ≤ /2, where if j = k , the corresponding sum is zero. 4. Show that n i=k+1 |b(ti ) − b(ti−1 )| = |b(x0 ) − b(tk )| ≤ /2. 5. Show that for all finite sequence t0 ≤ . . . ≤ tn , n ≥ 1, in [0, x0 ]: n |b(ti ) − b(ti−1 )| ≤ |b|(y0 ) − |b(0)| + i=1 6. Show that |b|(x0 ) ≤ |b|(y0 ) + . 7. Show that |b|(x0 ) ≤ |b|(x0 −) and that |b| is left-continuous. 8. Prove the following: Theorem 81 Let b : R+ → C be a map of finite variation. Then: b is right-continuous ⇒ |b| is right-continuous b is left-continuous ⇒ |b| is left-continuous b is continuous ⇒ |b| is continuous Tutorial 14: Maps of Finite Variation 13 Exercise 16. Let b : R+ → R be an R-valued map of finite variation. 1. Show that if b is right-continuous, then so are |b|+ and |b|− . 2. State and prove similar results for left-continuity and continuity. Exercise 17. Let b : R+ → C be a right-continuous map of finite variation. Show the existence of b1 , b2 , b3 , b4 : R+ → R+ , rightcontinuous and non-decreasing maps with bi (0) ≥ 0, such that: b = b1 − b2 + i(b3 − b4 ) Exercise 18. Let b : R+ → C be a right-continuous map. Let t ∈ R+ . For all p ≥ 1, we define: 2p Sp = |b(0)| + |b(kt/2p ) − b((k − 1)t/2p )| k=1 1. Show that for all p ≥ 1, Sp ≤ Sp+1 and define S = supp≥1 Sp . Tutorial 14: Maps of Finite Variation 14 2. Show that S ≤ |b|(t). Exercise 19. Further to exercise (18), let t0 < . . . < tn be a finite sequence of distinct elements of [0, t]. Let > 0. 1. Show that for all i = 0, . . . , n, there exists pi ≥ 1 and qi ∈ {0, 1, . . . , 2pi } such that: 0 ≤ t0 ≤ q0 t q1 t qn t < t1 ≤ p1 < . . . < tn ≤ pn ≤ t p0 2 2 2 and: |b(ti ) − b(qi t/2pi )| ≤ , ∀i = 0, . . . , n 2. Show the existence of p ≥ 1, and k0 , . . . , kn ∈ {0, . . . , 2p } with: 0 ≤ t0 ≤ k0 t k1 t kn t < t1 ≤ p < . . . < tn ≤ p ≤ t p 2 2 2 and: |b(ti ) − b(ki t/2p )| ≤ , ∀i = 0, . . . , n Tutorial 14: Maps of Finite Variation 3. Show that: n |b(ki t/2p ) − b(ki−1 t/2p )| ≤ Sp − |b(0)| i=1 4. Show that: n |b(ti ) − b(ti−1 )| ≤ S − |b(0)| + 2n i=1 5. Show that: n |b(ti ) − b(ti−1 )| ≤ S − |b(0)| i=1 6. Conclude that |b|(t) ≤ S . 7. Prove the following: 15 Tutorial 14: Maps of Finite Variation 16 Theorem 82 Let b : R+ → C be right-continuous or left-continuous. Then, for all t ∈ R+ : 2n |b(kt/2n ) − b((k − 1)t/2n )| |b|(t) = |b(0)| + lim n→+∞ k=1 Exercise 20. Let b : R+ → R+ be defined by b = 1Q+ . Show that: 2n |b(k/2n ) − b((k − 1)/2n )| = 0 +∞ = |b|(1) = lim n→+∞ k=1 Exercise 21. b : R+ → C is right-continuous of bounded variation. 1. Let b = b1 + ib2 . Explain why d|b1 |+ , d|b1 |− , d|b2 |+ and d|b2 |− are all well-defined measures on (R+ , B (R+ )). 2. Is this still true, if b is right-continuous of finite variation? 3. Show that d|b1 |+ , d|b1 |− , d|b2 |+ and d|b2 |− are finite measures. Tutorial 14: Maps of Finite Variation 17 4. Let db = d|b1 |+ − d|b1 |− + i(d|b2 |+ − d|b2 |− ). Show that db is a well-defined complex measure on (R+ , B (R+ )). 5. Show that db({0}) = b(0). 6. Show that for all s, t ∈ R+ , s ≤ t, db(]s, t]) = b(t) − b(s). 7. Show that limt→+∞ b(t) exists in C. We denote b(∞) this limit. 8. Show that db(R+ ) = b(∞). 9. Proving the uniqueness of db, justify the following: Definition 110 Let b : R+ → C be a right-continuous map of bounded variation. There exists a unique complex measure db on (R+ , B (R+ )), such that: (i) (ii) db({0}) = b(0) ∀s, t ∈ R+ s ≤ t , db(]s, t]) = b(t) − b(s) db is called the complex stieltjes measure associated with b. Tutorial 14: Maps of Finite Variation 18 Exercise 22. Show that if a : R+ → R+ is right-continuous, nondecreasing with a(0) ≥ 0 and a(∞) < +∞, then definition (110) of da coincide with the already known definition (24). Exercise 23. b : R+ → C is right-continuous of finite variation. 1. Let b = b1 + ib2 . Explain why d|b1 |+ , d|b1 |− , d|b2 |+ and d|b2 |− are all well-defined measures on (R+ , B (R+ )). 2. Why is it in general impossible to define: db = d|b1 |+ − d|b1 |− + i(d|b2 |+ − d|b2 |− ) Warning: it does not make sense to write something like ’db’, unless b is either right-continuous, non-decreasing and b(0) ≥ 0, or b is a right-continuous map of bounded variation. Exercise 24. Let b : R+ → C be a map. For all T ∈ R+ , we define bT : R+ → C as bT (t) = b(T ∧ t) for all t ∈ R+ . 1. Show that for all T ∈ R+ , |bT | = |b|T . Tutorial 14: Maps of Finite Variation 19 2. Show that if b is of finite variation, then for all T ∈ R+ , bT is of bounded variation, and we have |bT |(∞) = |b|(T ) < +∞. 3. Show that if b is right-continuous and of finite variation, for all T ∈ R+ , dbT is the unique complex measure on R+ , with: (i) (ii) dbT ({0}) = b(0) ∀s, t ∈ R+ , s ≤ t , dbT (]s, t]) = b(T ∧ t) − b(T ∧ s) 4. Show that if b is R-valued and of finite variation, for all T ∈ R+ , we have |bT |+ = (|b|+ )T and |bT |− = (|b|− )T . 5. Show that if b is right-continuous and of bounded variation, for all T ∈ R+ , we have dbT = db[0,T ] = db([0, T ] ∩ · ) 6. Show that if b is right-continuous, non-decreasing with b(0) ≥ 0, for all T ∈ R+ , we have dbT = db[0,T ] = db([0, T ] ∩ · ) Tutorial 14: Maps of Finite Variation 20 Exercise 25. Let µ, ν be two finite measures on R+ , such that: (i) (ii) µ({0}) ≤ ν ({0}) ∀s, t ∈ R+ , s ≤ t , µ(]s, t]) ≤ ν (]s, t]) We define a, c : R+ → R+ by a(t) = µ([0, t]) and c(t) = ν ([0, t]). 1. Show that a and c are right-continuous, non-decreasing with a(0) ≥ 0 and c(0) ≥ 0. 2. Show that da = µ and dc = ν . 3. Show that a ≤ c. 4. Define b : R+ → R+ by b = c − a. Show that b is rightcontinuous, non-decreasing with b(0) ≥ 0. 5. Show that da + db = dc. 6. Conclude with the following: Tutorial 14: Maps of Finite Variation 21 Theorem 83 Let µ, ν be two finite measures on (R+ , B (R+ )) with: (i) (ii) µ({0}) ≤ ν ({0}) ∀s, t ∈ R+ , s ≤ t , µ(]s, t]) ≤ ν (]s, t]) Then µ ≤ ν , i.e. for all B ∈ B (R+ ), µ(B ) ≤ ν (B ). Exercise 26. b : R+ → C is right-continuous of bounded variation. 1. Show that |db|({0}) = |b(0)| = d|b|({0}). Let s, t ∈ R+ , s ≤ t. 2. Let t0 ≤ . . . ≤ tn be a finite sequence in [s, t]. Show: n |b(ti ) − b(ti−1 )| ≤ |db|(]s, t]) i=1 3. Show that |b|(t) − |b|(s) ≤ |db|(]s, t]). 4. Show that d|b| ≤ |db|. Tutorial 14: Maps of Finite Variation 22 5. Show that L1 (R+ , B (R+ ), |db|) ⊆ L1 (R+ , B (R+ ), d|b|). C C 6. Show that R+ is metrizable and strongly σ -compact. c b 7. Show that CC (R+ ), CC (R+ ) are dense in L1 (R+ , B (R+ ), |db|). C 8. Let h ∈ L1 (R+ , B (R+ ), |db|). Given > 0, show the existence C b of φ ∈ CC (R+ ) such that |φ − h||db| ≤ . 9. Show that | hdb| ≤ | φdb| + . 10. Show that: |φ|d|b| − 11. Show that |h|d|b| ≤ |φ|d|b| ≤ |φ − h|d|b| ≤ |φ − h||db| |h|d|b| + . 12. For all n ≥ 1, we define: n2n −1 φ(k/2n )1]k/2n ,(k+1)/2n ] φn = φ(0)1{0} + k=0 Tutorial 14: Maps of Finite Variation 23 Show there is M ∈ R+ , such that |φn (x)| ≤ M for all x and n. 13. Using the continuity of φ, show that φn → φ. 14. Show that lim φn db = φdb. 15. Show that lim |φn |d|b| = |φ|d|b|. 16. Show that for all n ≥ 1: n2n −1 φ(k/2n )(b((k + 1)/2n ) − b(k/2n )) φn db = φ(0)b(0) + k=0 17. Show that | φn db| ≤ |φn |d|b| for all n ≥ 1. 18. Show that | φdb| ≤ |φ|d|b|. 19. Show that | hdb| ≤ |h|d|b| + 2 . 20. Show that | hdb| ≤ |h|d|b| for all h ∈ L1 (R+ , B (R+ ), |db|). C Tutorial 14: Maps of Finite Variation 24 21. Let B ∈ B (R+ ) and h ∈ L1 (R+ , B (R+ ), |db|) be such that C ¯ |h| = 1 and db = h|db|. Show that |db|(B ) = B hdb. 22. Conclude that |db| ≤ d|b|. Exercise 27. b : R+ → C is right-continuous of finite variation. 1. Show that for all T ∈ R+ , |dbT | = d|bT | and d|bT | = d|b|T . 2. Show that d|b|T = d|b|[0,T ] = d|b|([0, T ] ∩ · ), and conclude: Theorem 84 If b : R+ → C is right-continuous of bounded variation, the total variation of its associated complex stieltjes measure, is equal to the stieltjes measure associated with its total variation, i.e. |db| = d|b| If b : R+ → C is right-continuous of finite variation, then for all T ∈ R+ , bT defined by bT (t) = b(T ∧ t), is right-continuous of bounded variation, and we have |dbT | = d|b|([0, T ] ∩ · ). Tutorial 14: Maps of Finite Variation 25 Definition 111 Let b : R+ → E be a map, where E is a topological space. We say that b is cadlag with respect to E , if and only if b is right-continuous, and the limit: b(t−) = lim b(s) s↑↑t exists in E , for all t ∈ R \ {0}. In the case when E = C, given b cadlag, we define b(0−) = 0, and for all t ∈ R+ : + ∆b(t) = b(t) − b(t−) Exercise 28. Let b : R+ → E be cadlag, where E is a topological space. Suppose b has values in E ⊆ E . 1. Explain why b may not be cadlag with respect to E . ¯ 2. Show that b is cadlag with respect to E . 3. Show that b : R+ → R is cadlag ⇔ it is cadlag w.r. to C. Tutorial 14: Maps of Finite Variation 26 Exercise 29. 1. Show that if b : R+ → C is cadlag, then b is continuous with b(0) = 0 if and only if ∆b(t) = 0 for all t ∈ R+ . 2. Show that if a : R+ → R+ is right-continuous, non-decreasing with a(0) ≥ 0, then a is cadlag (w.r. to R) with ∆a ≥ 0. 3. Show that any linear combination of cadlag maps is itself cadlag. 4. Show that if b : R+ → C is a right-continuous map of finite variation, then b is cadlag. 5. Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Show that da({t}) = ∆a(t) for all t ∈ R+ . 6. Let b : R+ → C be a right-continuous map of bounded variation. Show that db({t}) = ∆b(t) for all t ∈ R+ . Tutorial 14: Maps of Finite Variation 27 7. Let b : R+ → C be a right-continuous map of finite variation. Let T ∈ R+ . Show that: ∀t ∈ R+ , bT (t−) = b(t−) b(T ) if if t≤T T <t Show that ∆bT = (∆b)1[0,T ] , and dbT ({t}) = ∆b(t)1[0,T ] (t). Exercise 30. Let b : R+ → C be a cadlag map and T ∈ R+ . 1. Show that if t → b(t−) is not bounded on [0, T ], there exists a sequence (tn )n≥1 in [0, T ] such that |b(tn )| → +∞. 2. Suppose from now on that b is not bounded on [0, T ]. Show the existence of a sequence (tn )n≥1 in [0, T ], such that tn → t for some t ∈ [0, T ], and |b(tn )| → +∞. 3. Define R = {n ≥ 1 : t ≤ tn } and L = {n ≥ 1 : tn < t}. Show that R and L cannot be both finite. Tutorial 14: Maps of Finite Variation 28 4. Suppose that R is infinite. Show the existence of n1 ≥ 1, with: tn1 ∈ [t, t + 1[∩[0, T ] 5. If R is infinite, show there is n1 < n2 < . . . such that: tnk ∈ [t, t + 1 [∩[0, T ] , ∀k ≥ 1 k 6. Show that |b(tnk )| → +∞. 7. Show that if L is infinite, then t > 0 and there is an increasing sequence n1 < n2 < . . ., such that: tnk ∈]t − 1 , t[∩[0, T ] , ∀k ≥ 1 k 8. Show that: |b(tnk )| → +∞. 9. Prove the following: Tutorial 14: Maps of Finite Variation 29 Theorem 85 Let b : R+ → C be a cadlag map. Let T ∈ R+ . Then b and t → b(t−) are bounded on [0, T ], i.e. there exists M ∈ R+ such that: |b(t)| ∨ |b(t−)| ≤ M , ∀t ∈ [0, T ] Tutorial 15: Stieltjes Integration 1 15. Stieltjes Integration Definition 112 b : R+ → C is right-continuous of finite variation. The stieltjes L1 -spaces associated with b are defined as: L1 (b) = C f : R+ → C measurable, L1,loc (b) = C |f |d|b| < +∞ f : R+ → C measurable, t |f |d|b| < +∞, ∀t ∈ R+ 0 t Warning : In these tutorials, 0 . . . refers to [0,t] . . ., i.e. the domain of integration is always [0, t], not ]0, t], [0, t[, or ]0, t[. Exercise 1. b : R+ → C is right-continuous of finite variation. 1. Propose a definition for L1 (b) and L1,loc (b). R R 2. Is L1 (b) the same thing as L1 (R+ , B (R+ ), d|b|)? C C 3. Is it meaningful to speak of L1 (R+ , B (R+), |db|)? C Tutorial 15: Stieltjes Integration 2 4. Show that L1 (b) = L1 (|b|) and L1,loc (b) = L1,loc (|b|). C C C C 5. Show that L1 (b) ⊆ L1,loc (b). C C Exercise 2. Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. For all f ∈ L1,loc (a), we define f.a : R+ → C as: C t f da , ∀t ∈ R+ f.a(t) = 0 1. Explain why f.a : R+ → C is a well-defined map. 2. Let t ∈ R+ , (tn )n≥1 be a sequence in R+ with tn ↓↓ t. Show: lim n→+∞ f 1[0,tn ] da = 3. Show that f.a is right-continuous. f 1[0,t] da Tutorial 15: Stieltjes Integration 3 4. Let t ∈ R+ and t0 ≤ . . . ≤ tn be a finite sequence in [0, t]. Show: n |f.a(ti ) − f.a(ti−1 )| ≤ |f |da0,t] i=1 5. Show that f.a is a map of finite variation with: t |f |da , ∀t ∈ R+ |f.a|(t) ≤ 0 Exercise 3. Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Let f ∈ L1 (a). C 1. Show that f.a is a right-continuous map of bounded variation. 2. Show d(f.a)([0, t]) = ν ([0, t]), for all t ∈ R+ , where ν = 3. Prove the following: f da. Tutorial 15: Stieltjes Integration 4 Theorem 86 Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Let f ∈ L1 (a). The map f.a : R+ → C defined by: C t f da , ∀t ∈ R+ f.a(t) = 0 is a right-continuous map of bounded variation, and its associated complex stieltjes measure is given by d(f.a) = f da, i.e. f da , ∀B ∈ B (R+ ) d(f.a)(B ) = B Exercise 4. Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Let f ∈ L1,loc (a), f ≥ 0. R 1. Show f.a is right-continuous, non-decreasing with f.a(0) ≥ 0. 2. Show d(f.a)([0, t]) = µ([0, t]), for all t ∈ R+ , where µ = f da. 3. Prove that d(f.a)([0, T ] ∩ · ) = µ([0, T ] ∩ · ), for all T ∈ R+ . Tutorial 15: Stieltjes Integration 5 4. Prove with the following: Theorem 87 Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Let f ∈ L1,loc (a), f ≥ 0. The map f.a : R+ → R+ R defined by: t f da , ∀t ∈ R+ f.a(t) = 0 is right-continuous, non-decreasing with (f.a)(0) ≥ 0, and its associated stieltjes measure is given by d(f.a) = f da, i.e. f da , ∀B ∈ B (R+ ) d(f.a)(B ) = B Exercise 5. Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Let f ∈ L1,loc (a) and T ∈ R+ . C 1. Show that |f |1[0,T ] da = |f |da[0,T ] = |f |daT . Tutorial 15: Stieltjes Integration 6 2. Show that f 1[0,T ] ∈ L1 (a) and f ∈ L1 (aT ). C C 3. Show that (f.a)T = f.(aT ) = (f 1[0,T ] ).a. 4. Show that for all B ∈ B (R+ ): d(f.a)T (B ) = f daT = B f 1[0,T ] da B 5. Explain why it does not in general make sense to write: d(f.a)T = d(f.a)([0, T ] ∩ · ) 6. Show that for all B ∈ B (R+ ): |d(f.a)T |(B ) = |f |1[0,T ] da , ∀B ∈ B (R+ ) B 7. Show that |d(f.a)T | = d|f.a|([0, T ] ∩ · ) Tutorial 15: Stieltjes Integration 7 8. Show that for all t ∈ R+ t |f.a|(t) = (|f |.a)(t) = |f |da 0 9. Show that f.a is of bounded variation if and only if f ∈ L1 (a). C 10. Show that ∆(f.a)(0) = f (0)∆a(0). 11. Let t > 0, (tn )n≥1 be a sequence in R+ with tn ↑↑ t. Show: lim n→+∞ f 1[0,tn ] da = f 1[0,t[ da 12. Show that ∆(f.a)(t) = f (t)∆a(t) for all t ∈ R+ . 13. Show that if a is continuous with a(0) = 0, then f.a is itself continuous with (f.a)(0) = 0. 14. Prove with the following: Tutorial 15: Stieltjes Integration 8 Theorem 88 Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Let f ∈ L1,loc (a). The map f.a : R+ → C defined by: C t f da , ∀t ∈ R+ f.a(t) = 0 is right-continuous of finite variation, and we have |f.a| = |f |.a, i.e. t |f |da , ∀t ∈ R+ |f.a|(t) = 0 In particular, f.a is of bounded variation if and only if f ∈ L1 (a). C Furthermore, we have ∆(f.a) = f ∆a. Exercise 6. Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Let b : R+ → C be right-continuous of finite variation. 1. Prove the equivalence between the following: (i) d|b| << da Tutorial 15: Stieltjes Integration (ii) (iii) 9 |dbT | << da , ∀T ∈ R+ dbT << da , ∀T ∈ R+ 2. Does it make sense in general to write db << da? Definition 113 Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Let b : R+ → C be right-continuous of finite variation. We say that b is absolutely continuous with respect to a, and we write b << a, if and only if, one of the following holds: (i) (ii) (iii) d|b| << da |dbT | << da , ∀T ∈ R+ dbT << da , ∀T ∈ R+ In other words, b is absolutely continuous w.r. to a, if and only if the stieltjes measure associated with the total variation of b is absolutely continuous w.r. to the stieltjes measure associated with a. Tutorial 15: Stieltjes Integration 10 Exercise 7. Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Let b : R+ → C be right-continuous of finite variation, absolutely continuous w.r. to a, i.e. with b << a. 1. Show that for all T ∈ R+ , there exits fT ∈ L1 (a) such that: C fT da , ∀B ∈ B (R+ ) dbT (B ) = B 2. Suppose that T, T ∈ R fT da = B + and T ≤ T . Show that: B ∩[0,T ] fT da , ∀B ∈ B (R+ ) 3. Show that fT = fT 1[0,T ] da-a.s. 4. Show the existence of a sequence (fn )n≥1 in L1 (a), such that C for all 1 ≤ n ≤ p, fn = fp 1[0,n] and: fn da , ∀B ∈ B (R+ ) ∀n ≥ 1 , dbn (B ) = B Tutorial 15: Stieltjes Integration 11 5. We define f : (R+ , B (R+ )) → (C, B (C)) by: ∀t ∈ R+ , f (t) = fn (t) for any n ≥ 1 : t ∈ [0, n] Explain why f is unambiguously defined. ∞ 6. Show that for all B ∈ B (C), {f ∈ B } = ∪+=1 [0, n] ∩ {fn ∈ B }. n 7. Show that f : (R+ , B (R+ )) → (C, B (C)) is measurable. 8. Show that f ∈ L1,loc (a) and that we have: C t f da , ∀t ∈ R+ b(t) = 0 9. Prove the following: Tutorial 15: Stieltjes Integration 12 Theorem 89 Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Let b : R+ → C be a right-continuous map of finite variation. Then, b is absolutely continuous w.r. to a, i.e. d|b| << da, if and only if there exists f ∈ L1,loc (a) such that b = f.a, i.e. C t f da , ∀t ∈ R+ b(t) = 0 If b is R-valued, we can assume that f ∈ L1,loc (a). R If b is non-decreasing with b(0) ≥ 0, we can assume that f ≥ 0. Exercise 8. Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. Let f, g ∈ L1,loc (a) be such that f.a = g.a, i.e.: C t t gda , ∀t ∈ R+ f da = 0 0 Tutorial 15: Stieltjes Integration 13 1. Show that for all T ∈ R+ and B ∈ B (R+ ): d(f.a)T (B ) = f 1[0,T ] da = B g 1[0,T ] da B 2. Show that for all T ∈ R+ , f 1[0,T ] = g 1[0,T ] da-a.s. 3. Show that f = g da-a.s. Exercise 9. b : R+ → C is right-continuous of finite variation. 1. Show the existence of h ∈ L1,loc (|b|) such that b = h.|b|. C 2. Show that for all B ∈ B (R+ ) and T ∈ R+ : hd|b|T = dbT (B ) = B h|dbT | B 3. Show that |h| = 1 |dbT |-a.s. for all T ∈ R+ . 4. Show that for all T ∈ R+ , d|b|([0, T ] ∩ {|h| = 1}) = 0. Tutorial 15: Stieltjes Integration 14 5. Show that |h| = 1 d|b|-a.s. 6. Prove the following: Theorem 90 Let b : R+ → C be right-continuous of finite variation. There exists h ∈ L1,loc (|b|) such that |h| = 1 and b = h.|b|, i.e. C t hd|b| , ∀t ∈ R+ b(t) = 0 Definition 114 b : R+ → C is right-continuous of finite variation. For all f ∈ L1 (b), the stieltjes integral of f with respect to b, is C defined as: f db = f hd|b| where h ∈ L1,loc (|b|) is such that |h| = 1 and b = h.|b|. C Tutorial 15: Stieltjes Integration 15 Warning : the notation f db of definition (114) is controversial and potentially confusing: ’db’ is not in general a complex measure on R+ , unless b is of bounded variation. Exercise 10. b : R+ → C is right-continuous of finite variation. 1. Show that if f ∈ L1 (b), then C 2. Explain why, given f ∈ L1 (b), C f hd|b| is well-defined. f db is unambiguously defined. 3. Show that if b is right-continuous, non-decreasing with b(0) ≥ 0, definition (114) of f db coincides with that of an integral w.r. to the stieltjes measure db. 4. Show that if b is a right-continuous map of bounded variation, definition (114) of f db coincides with that of an integral w.r. to the complex stieltjes measure db. Tutorial 15: Stieltjes Integration 16 Exercise 11. Let b : R+ → C be a right-continuous map of finite variation. For all f ∈ L1,loc (b), we define f.b : R+ → C as: C t f db , ∀t ∈ R+ f.b(t) = 0 1. Explain why f.b : R+ → C is a well-defined map. 2. If b is right-continuous, non-decreasing with b(0) ≥ 0, show this definition of f.b coincides with that of theorem (88). 3. Show f.b = (f h).|b|, where h ∈ L1,loc (|b|), |h| = 1, b = h.|b|. C 4. Show that f.b : R+ → C is right-continuous of finite variation, with |f.b| = |f |.|b|, i.e. t |f |d|b| , ∀t ∈ R+ |f.b|(t) = 0 5. Show that f.b is of bounded variation if and only if f ∈ L1 (b). C Tutorial 15: Stieltjes Integration 17 6. Let t > 0, (tn )n≥1 be a sequence in R+ such that tn ↑↑ t. Show: lim n→+∞ f h1[0,tn ] d|b| = f h1[0,t[ d|b| 7. Show that ∆(f.b) = f ∆b. 8. Show that if b is continuous with b(0) = 0, then f.b is itself continuous with (f.b)(0) = 0. 9. Prove the following: Tutorial 15: Stieltjes Integration 18 Theorem 91 Let b : R+ → C be right-continuous of finite variation. For all f ∈ L1,loc (b), the map f.b : R+ → C defined by: C t f db , ∀t ∈ R+ f.b(t) = 0 is right-continuous of finite variation, and we have |f.b| = |f |.|b|, i.e. t |f |d|b| , ∀t ∈ R+ |f.b|(t) = 0 In particular, f.b is of bounded variation if and only if f ∈ L1 (b). C Furthermore, we have ∆(f.b) = f ∆b. Exercise 12. Let b : R+ → C be right-continuous of finite variation. Let f ∈ L1,loc (b) and T ∈ R+ . C 1. Show that |f |1[0,T ] d|b| = |f |d|b|[0,T ] = 2. Show that f 1[0,T ] ∈ L1 (b) and f ∈ L1 (bT ). C C |f |d|bT |. Tutorial 15: Stieltjes Integration 19 3. Show bT = h.|bT |, where h ∈ L1,loc (|b|), |h| = 1, b = h.|b|. C 4. Show that (f.b)T = f.(bT ) = (f 1[0,T ] ).b 5. Show that d|f.b|(B ) = B |f |d|b| for all B ∈ B (R+ ). 6. Let g : R+ → C be a measurable map. Show the equivalence: g ∈ L1,loc (f.b) ⇔ gf ∈ L1,loc (b) C C 7. Show that d(f.b)T (B ) = 8. Show that dbT = B f hd|bT | for all B ∈ B (R+ ). hd|bT | and conclude that: f dbT , ∀B ∈ B (R+ ) d(f.b)T (B ) = B 9. Let g ∈ L1,loc (f.b). Show that g ∈ L1 ((f.b)T ) and: C C g 1[0,t] d(f.b)T = g f 1[0,t]dbT , ∀t ∈ R+ Tutorial 15: Stieltjes Integration 20 10. Show that g. (f.b)T = (gf ).(bT ). 11. Show that (g.(f.b))T = ((gf ).b)T . 12. Show that g.(f.b) = (gf ).b 13. Prove the following: Theorem 92 Let b : R+ → C be right-continuous of finite variation. For all f ∈ L1,loc (b) and g : (R+ , B (R+ )) → (C, B (C)) measurable C map, we have the equivalence: g ∈ L1,loc (f.b) ⇔ gf ∈ L1,loc (b) C C and when such condition is satisfied, g.(f.b) = (f g ).b, i.e. t t gf db , ∀t ∈ R+ gd(f.b) = 0 0 Tutorial 15: Stieltjes Integration 21 Exercise 13. Let b : R+ → C be right-continuous of finite variation. let f, g ∈ L1,loc (b) and α ∈ C. Show that f + αg ∈ L1,loc (b), and: C C (f + αg ).b = f.b + αg.b Exercise 14. Let b, c : R+ → C be two right-continuous maps of finite variations. Let f ∈ L1,loc (b) ∩ L1,loc (c) and α ∈ C. C C 1. Show that for all T ∈ R+ , d(b + αc)T = dbT + αdcT . 2. Show that for all T ∈ R+ , d|b + αc|T ≤ d|b|T + |α|d|c|T . 3. Show that d|b + αc| ≤ d|b| + |α|d|c|. 4. Show that f ∈ L1,loc (b + αc). C 5. Show d(f.(b + αc))T (B ) = B f d(b + αc)T for all B ∈ B (R+ ). 6. Show that d(f.(b + αc))T = d(f.b)T + αd(f.c)T . Tutorial 15: Stieltjes Integration 22 7. Show that (f.(b + αc))T = (f.b)T + α(f.c)T 8. Show that f.(b + αc) = f.b + α(f.c). Exercise 15. Let b : R+ → C be right-continuous of finite variation. 1. Show that d|b| ≤ d|b1 | + d|b2 |, where b = b1 + ib2 . 2. Show that d|b1 | ≤ d|b| and d|b2 | ≤ d|b|. 3. Show that f ∈ L1,loc (b), if and only if: C f ∈ L1,loc (|b1 |+ ) ∩ L1,loc (|b1 |− ) ∩ L1,loc (|b2 |+ ) ∩ L1,loc (|b2 |− ) C C C C 4. Show that if f ∈ L1,loc (b), for all t ∈ R+ : C t t 0 t f d|b1 |+ − f db = 0 0 t f d|b1 |− + i t f d|b2 |+ − 0 0 f d|b2 |− Tutorial 15: Stieltjes Integration 23 Exercise 16. Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. We define c : R+ → [0, +∞] as: c(t) = inf {s ∈ R+ : t < a(s)} , ∀t ∈ R+ where it is understood that inf ∅ = +∞. Let s, t ∈ R+ . 1. Show that t < a(s) ⇒ c(t) ≤ s. 2. Show that c(t) < s ⇒ t < a(s). 3. Show that c(t) ≤ s ⇒ t < a(s + ) , ∀ > 0. 4. Show that c(t) ≤ s ⇒ t ≤ a(s). 5. Show that c(t) < +∞ ⇔ t < a(∞). 6. Show that c is non-decreasing. 7. Show that if t0 ∈ [a(∞), +∞[, c is right-continuous at t0 . Tutorial 15: Stieltjes Integration 24 8. Suppose t0 ∈ [0, a(∞)[. Given > 0, show the existence of s ∈ R+ , such that c(t0 ) ≤ s < c(t0 ) + and t0 < a(s). 9. Show that t ∈ [t0 , a(s)[ ⇒ c(t0 ) ≤ c(t) ≤ c(t0 ) + . 10. Show that c is right-continuous. 11. Show that if a(∞) = +∞, then c is a map c : R+ → R+ which is right-continuous, non-decreasing with c(0) ≥ 0. 12. We define a(s) = inf {t ∈ R+ : s < c(t)} for all s ∈ R+ . Show ¯ that for all s, t ∈ R+ , s < c(t) ⇒ a(s) ≤ t. 13. Show that a ≤ a. ¯ 14. Show that for all s, t ∈ R+ and > 0: a(s + ) ≤ t ⇒ s < s + ≤ c(t) ¯ 15. Show that for all s, t ∈ R+ and > 0, a(s + ) ≤ t ⇒ a(s) ≤ t. Tutorial 15: Stieltjes Integration 25 16. Show that a ≤ a and conclude that: ¯ a(s) = inf {t ∈ R+ : s < c(t)} ¯ Exercise 17. Let f : R+ → R be a non-decreasing map. Let α ∈ R. We define: x0 = sup{x ∈ R+ : f (x) ≤ α} 1. Explain why x0 = −∞ if and only if {f ≤ α} = ∅. 2. Show that x0 = +∞ if and only if {f ≤ α} = R+ . 3. We assume from now on that x0 = ±∞. Show that x0 ∈ R+ . 4. Show that if f (x0 ) ≤ α then {f ≤ α} = [0, x0 ]. 5. Show that if α < f (x0 ) then {f ≤ α} = [0, x0 [. ¯ ¯ 6. Conclude that f : (R+ , B (R+ )) → (R, B (R)) is measurable. Tutorial 15: Stieltjes Integration 26 Exercise 18. Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. We define c : R+ → [0, +∞] as: c(t) = inf {s ∈ R+ : t < a(s)} , ∀t ∈ R+ 1. Let f : R+ → [0, +∞] be non-negative and measurable. Show (f ◦ c)1{c<+∞} is well-defined, non-negative and measurable. 2. Let t, u ∈ R+ , and ds be the lebesgue measure on R+ . Show: a(t) (1[0,u] ◦ c)1{c<+∞} ds ≤ 1[0,a(t∧u)] 1{c<+∞} ds 0 3. Show that: a(t) (1[0,u] ◦ c)1{c<+∞} ds ≤ a(t ∧ u) 0 4. Show that: a(t) a(t ∧ u) = a(t) 1[0,a(u)[ ds = 0 0 1[0,a(u)[ 1{c<+∞} ds Tutorial 15: Stieltjes Integration 27 5. Show that: a(t) (1[0,u] ◦ c)1{c<+∞} ds a(t ∧ u) ≤ 0 6. Show that: t a(t) 1[0,u] da = 0 0 (1[0,u] ◦ c)1{c<+∞} ds 7. Define: t Dt = B ∈ B (R+ ) : a(t) (1B ◦ c)1{c<+∞} ds 1B da = 0 0 Show that Dt is a dynkin system on R+ , and Dt = B (R+ ). 8. Show that if f : R+ → [0, +∞] is non-negative measurable: t a(t) (f ◦ c)1{c<+∞} ds , ∀t ∈ R+ f da = 0 0 Tutorial 15: Stieltjes Integration 28 9. Let f : R+ → C be measurable. Show that (f ◦ c)1{c<+∞} is itself well-defined and measurable. 10. Show that if f ∈ L1,loc (a), then for all t ∈ R+ , we have: C (f ◦ c)1{c<+∞} 1[0,a(t)] ∈ L1 (R+ , B (R+ ), ds) C and furthermore: t a(t) (f ◦ c)1{c<+∞} ds f da = 0 0 11. Show that we also have: t f da = 0 12. Conclude with the following: (f ◦ c)1[0,a(t)[ ds Tutorial 15: Stieltjes Integration 29 Theorem 93 Let a : R+ → R+ be right-continuous, non-decreasing with a(0) ≥ 0. We define c : R+ → [0, +∞] as: c(t) = inf {s ∈ R+ : t < a(s)} , ∀t ∈ R+ Then, for all f ∈ L1,loc (a), we have: C t a(t) (f ◦ c)1{c<+∞} ds , ∀t ∈ R+ f da = 0 0 where ds is the lebesgue measure on R+ . Tutorial 16: Differentiation 1 16. Differentiation ¯ Definition 115 Let (Ω, T ) be a topological space. A map f : Ω → R is said to be lower-semi-continuous (l.s.c), if and only if: ∀λ ∈ R , {λ < f } is open We say that f is upper-semi-continuous (u.s.c), if and only if: ∀λ ∈ R , {f < λ} is open ¯ Exercise 1. Let f : Ω → R be a map, where Ω is a topological space. ¯ 1. Show that f is l.s.c if and only if {λ < f } is open for all λ ∈ R. ¯ 2. Show that f is u.s.c if and only if {f < λ} is open for all λ ∈ R. ¯ 3. Show that every open set U in R can be written: U =V+∪V−∪ αi , βi [ i∈I Tutorial 16: Differentiation 2 for some index set I , αi , βi ∈ R, V + = ∅ or V + =]α, +∞], (α ∈ R) and V − = ∅ or V − = [−∞, β [, (β ∈ R). 4. Show that f is continuous if and only if it is both l.s.c and u.s.c. ¯ 5. Let u : Ω → R and v : Ω → R. Let λ ∈ R. Show that: {λ1 < u} ∩ {λ2 < v } {λ < u + v } = (λ1 , λ2 ) ∈ R λ1 + λ2 = λ 2 6. Show that if both u and v are l.s.c, then u + v is also l.s.c. 7. Show that if both u and v are u.s.c, then u + v is also u.s.c. 8. Show that if f is l.s.c, then αf is l.s.c, for all α ∈ R+ . 9. Show that if f is u.s.c, then αf is u.s.c, for all α ∈ R+ . 10. Show that if f is l.s.c, then −f is u.s.c. Tutorial 16: Differentiation 3 11. Show that if f is u.s.c, then −f is l.s.c. 12. Show that if V is open in Ω, then f = 1V is l.s.c. 13. Show that if F is closed in Ω, then f = 1F is u.s.c. ¯ Exercise 2. Let (fi )i∈I be an arbitrary family of maps fi : Ω → R, defined on a topological space Ω. 1. Show that if all fi ’s are l.s.c, then f = supi∈I fi is l.s.c. 2. Show that if all fi ’s are u.s.c, then f = inf i∈I fi is u.s.c. Exercise 3. Let (Ω, T ) be a metrizable and σ -compact topological space. Let µ be a locally finite measure on (Ω, B (Ω)). Let f be an element of ∈ L1 (Ω, B (Ω), µ), such that f ≥ 0. R 1. Let (sn )n≥1 be a sequence of simple functions on (Ω, B (Ω)) such that sn ↑ f . Define t1 = s1 and tn = sn − sn−1 for all n ≥ 2. Show that tn is a simple function on (Ω, B (Ω)), for all n ≥ 1. Tutorial 16: Differentiation 4 2. Show that f can be written as: +∞ αn 1An f= n=1 where αn ∈ R+ \ {0} and An ∈ B (Ω), for all n ≥ 1. 3. Show that µ(An ) < +∞, for all n ≥ 1. 4. Show that there exist Kn compact and Vn open in Ω such that: Kn ⊆ An ⊆ Vn , µ(Vn \ Kn ) ≤ for all > 0 and n ≥ 1. 5. Show the existence of N ≥ 1 such that: +∞ αn µ(An ) ≤ n=N +1 2 αn 2n+1 Tutorial 16: Differentiation 5 6. Define u = N n=1 αn 1Kn . Show that u is u.s.c. 7. Define v = +∞ n=1 αn 1Vn . Show that v is l.s.c. 8. Show that we have 0 ≤ u ≤ f ≤ v . 9. Show that we have: +∞ +∞ αn 1Kn + v =u+ n=N +1 10. Show that v dµ ≤ αn 1Vn \Kn n=1 udµ + < +∞. 11. Show that u ∈ L1 (Ω, B (Ω), µ). R 12. Explain why v may fail to be in L1 (Ω, B (Ω), µ). R 13. Show that v is µ-a.s. equal to an element of L1 (Ω, B (Ω), µ). R 14. Show that (v − u)dµ ≤ . Tutorial 16: Differentiation 6 15. Prove the following: Theorem 94 (Vitali-Caratheodory) Let (Ω, T ) be a metrizable and σ -compact topological space. Let µ be a locally finite measure on (Ω, B (Ω)) and f be an element of L1 (Ω, B (Ω), µ). Then, for all R ¯ > 0, there exist maps u, v : Ω → R, which are µ-a.s. equal to elements of L1 (Ω, B (Ω), µ), such that u ≤ f ≤ v , u is u.s.c, v is l.s.c, R and furthermore: (v − u)dµ ≤ Definition 116 We call connected topological space, a topological space (Ω, T ), for which the only subsets of Ω which are both open and closed, are Ω and ∅. Tutorial 16: Differentiation 7 Exercise 4. Let (Ω, T ) be a topological space. 1. Show that (Ω, T ) is connected if and only if whenever Ω = A B where A, B are disjoint open sets, we have A = ∅ or B = ∅. 2. Show that (Ω, T ) is connected if and only if whenever Ω = A B where A, B are disjoint closed sets, we have A = ∅ or B = ∅. Definition 117 Let (Ω, T ) be a topological space, and A ⊆ Ω. We say that A is a connected subset of Ω, if and only if the induced topological space (A, T|A ) is connected. Exercise 5. Let A be open and closed in R, with A = ∅ and Ac = ∅. 1. Let x ∈ Ac . Show that A ∩ [x, +∞[ or A∩] − ∞, x] is non-empty. 2. Suppose B = A ∩ [x, +∞[= ∅. Show that B is closed and that we have B = A∩]x, +∞[. Conclude that B is also open. Tutorial 16: Differentiation 8 3. Let b = inf B . Show that b ∈ B (and in particular b ∈ R). 4. Show the existence of > 0 such that ]b − , b + [⊆ B . 5. Conclude with the following: Theorem 95 The topological space (R, TR ) is connected. Exercise 6. Let (Ω, T ) be a topological space and A ⊆ Ω be a ¯ connected subset of Ω. Let B be a subset of Ω such that A ⊆ B ⊆ A. We assume that B = V1 V2 where V1 , V2 are disjoint open sets in B . 1. Show there is U1 , U2 open in Ω, with V1 = B ∩ U1 , V2 = B ∩ U2 . 2. Show that A ∩ U1 = ∅ or A ∩ U2 = ∅. c ¯ 3. Suppose that A ∩ U1 = ∅. Show that A ⊆ U1 . 4. Show then that V1 = B ∩ U1 = ∅. Tutorial 16: Differentiation 9 ¯ 5. Conclude that B and A are both connected subsets of Ω. Exercise 7. Prove the following: Theorem 96 Let (Ω, T ), (Ω , T ) be two topological spaces, and f be a continuous map, f : Ω → Ω . If (Ω, T ) is connected, then f (Ω) is a connected subset of Ω . ¯ Definition 118 Let A ⊆ R. We say that A is an interval, if and only if for all x, y ∈ A with x ≤ y , we have [x, y ] ⊆ A, where: ¯ [x, y ] = {z ∈ R : x ≤ z ≤ y } ¯ Exercise 8. Let A ⊆ R. 1. If A is an interval, and α = inf A, β = sup A, show that:α, β [⊆ A ⊆ [α, β ] Tutorial 16: Differentiation 10 2. Show that A is an interval if and only if, it is of the form [α, β ], ¯ [α, β [, ]α, β ] or ]α, β [, for some α, β ∈ R. 3. Show that an interval of the form ] − ∞, α[, where α ∈ R, is homeomorphic to ] − 1, α [, for some α ∈ R. 4. Show that an interval of the form ]α, +∞[, where α ∈ R, is homeomorphic to ]α , 1[, for some α ∈ R. 5. Show that an interval of the form ]α, β [, where α, β ∈ R and α < β , is homeomorphic to ] − 1, 1[. 6. Show that ] − 1, 1[ is homeomorphic to R. 7. Show an non-empty open interval in R, is homeomorphic to R. 8. Show that an open interval in R, is a connected subset of R. 9. Show that an interval in R, is a connected subset of R. Tutorial 16: Differentiation 11 Exercise 9. Let A ⊆ R be a non-empty connected subset of R, and α = inf A, β = sup A. We assume there exists x0 ∈ Ac ∩]α, β [. 1. Show that A∩]x0 , +∞[ or A∩] − ∞, x0 [ is empty. 2. Show that if A∩]x0 , +∞[= ∅, then β cannot be sup A. 3. Show that ]α, β [⊆ A ⊆ [α, β ]. 4. Show the following: Theorem 97 For all A ⊆ R, A is a connected subset of R , if and only if A is an interval. Exercise 10. Prove the following: Theorem 98 Let f : Ω → R be a continuous map, where (Ω, T ) is a connected topological space. Let a, b ∈ Ω such that f (a) ≤ f (b). Then, for all z ∈ [f (a), f (b)], there exists x ∈ Ω such that z = f (x). Tutorial 16: Differentiation 12 Exercise 11. Let a, b ∈ R, a < b, and f : [a, b] → R be a map such that f (x) exists for all x ∈ [a, b]. 1. Show that f : ([a, b], B ([a, b])) → (R, B (R)) is measurable. 2. Show that f ∈ L1 ([a, b], B ([a, b]), dx) is equivalent to: R b |f (t)|dt < +∞ a 3. We assume from now on that f ∈ L1 ([a, b], B ([a, b]), dx). Given R ¯ > 0, show the existence of g : [a, b] → R, almost surely equal 1 to an element of LR ([a, b], B ([a, b]), dx), such that f ≤ g and g is l.s.c, with: b b g (t)dt ≤ a f (t)dt + a 4. By considering g + α for some α > 0, show that without loss of generality, we can assume that f < g with the above inequality still holding. Tutorial 16: Differentiation 13 5. We define the complex measure ν = Show that: g dx ∈ M 1 ([a, b], B ([a, b])). ∀ > 0 , ∃δ > 0 , ∀E ∈ B ([a, b]) , dx(E ) ≤ δ ⇒ |ν (E )| < 6. For all η > 0 and x ∈ [a, b], we define: x g (t)dt − f (x) + f (a) + η (x − a) Fη (x) = a Show that Fη : [a, b] → R is a continuous map. − 7. η being fixed, let x = sup Fη 1 ({0}). Show that x ∈ [a, b] and Fη (x) = 0. 8. We assume that x ∈ [a, b[. Show the existence of δ > 0 such that for all t ∈]x, x + δ [∩[a, b], we have: f (x) < g (t) and f (t) − f (x) < f (x) + η t−x Tutorial 16: Differentiation 14 9. Show that for all t ∈]x, x + δ [∩[a, b], we have Fη (t) > Fη (x) = 0. 10. Show that there exists t0 such that x < t0 < b and Fη (t0 ) > 0. − 11. Show that if Fη (b) < 0 then x cannot be sup Fη 1 ({0}). 12. Conclude that Fη (b) ≥ 0, even if x = b. 13. Show that f (b) − f (a) ≤ b a f (t)dt, and conclude: Theorem 99 (Fundamental Calculus) Let a, b ∈ R, a < b, and f : [a, b] → R be a map which is differentiable at every point of [a, b], and such that: b |f (t)|dt < +∞ a Then, we have: b f (b) − f (a) = f (t)dt a Tutorial 16: Differentiation 15 Exercise 12. Let α > 0, and kα : Rn → Rn defined by kα (x) = αx. 1. Show that kα : (Rn , B (Rn )) → (Rn , B (Rn )) is measurable. 2. Show that for all B ∈ B (Rn ), we have: dx({kα ∈ B }) = 3. Show that for all 1 dx(B ) αn > 0 and x ∈ Rn : dx(B (x, )) = n dx(B (0, 1)) Definition 119 Let µ be a complex measure on (Rn , B (Rn )), n ≥ 1, with total variation |µ|. We call maximal function of µ, the map M µ : Rn → [0, +∞], defined by: ∀x ∈ Rn , (M µ)(x) = sup >0 n |µ|(B (x, )) dx(B (x, )) where B (x, ) is the open ball in R , of center x and radius , with respect to the usual metric of Rn . Tutorial 16: Differentiation 16 Exercise 13. Let µ be a complex measure on (Rn , B (Rn )). 1. Let λ ∈ R. Show that if λ < 0, then {λ < M µ} = Rn . 2. Show that if λ = 0, then {λ < M µ} = Rn if µ = 0, and {λ < M µ} is the empty set if µ = 0. 3. Suppose λ > 0. Let x ∈ {λ < M µ}. Show the existence of > 0 such that |µ|(B (x, )) = tdx(B (x, )), for some t > λ. 4. Show the existence of δ > 0 such that ( + δ )n < n t/λ. 5. Show that if y ∈ B (x, δ ), then B (x, ) ⊆ B (y, + δ ). 6. Show that if y ∈ B (x, δ ), then: |µ|(B (y, + δ )) ≥ n t dx(B (y, + δ )) > λdx(B (y, + δ )) ( + δ )n 7. Conclude that B (x, δ ) ⊆ {λ < M µ}, and that the maximal function M µ : Rn → [0, +∞] is l.s.c, and therefore measurable. Tutorial 16: Differentiation 17 Exercise 14. Let Bi = B (xi , i ), i = 1, . . . , N , N ≥ 1, be a finite collection of open balls in Rn . Assume without loss of generality that N ≤ . . . ≤ 1 . We define a sequence (Jk ) of sets by J0 = {1, . . . , N } and for all k ≥ 1: Jk = Jk−1 ∩ {j : j > ik , Bj ∩ Bik = ∅} ∅ if Jk−1 = ∅ if Jk−1 = ∅ where we have put ik = min Jk−1 , whenever Jk−1 = ∅. 1. Show that if Jk−1 = ∅ then Jk ⊂ Jk−1 (strict inclusion), k ≥ 1. 2. Let p = min{k ≥ 1 : Jk = ∅}. Show that p is well-defined. 3. Let S = {i1 , . . . , ip }. Explain why S is well defined. 4. Suppose that 1 ≤ k < k ≤ p. Show that ik ∈ Jk . 5. Show that (Bi )i∈S is a family of pairwise disjoint open balls. 6. Let i ∈ {1, . . . , N } \ S , and define k0 to be the minimum of the set {k ∈ Np : i ∈ Jk }. Explain why k0 is well-defined. Tutorial 16: Differentiation 18 7. Show that i ∈ Jk0 −1 and ik0 ≤ i. 8. Show that Bi ∩ Bik0 = ∅. 9. Show that Bi ⊆ B (xik0 , 3 ik0 ). 10. Conclude that there exists a subset S of {1, . . . , N } such that (Bi )i∈S is a family of pairwise disjoint balls, and: N B (xi , i ) ⊆ i=1 B (xi , 3 i ) i∈S 11. Show that: N B (xi , i ) dx i=1 ≤ 3n dx(B (xi , i )) i∈S Tutorial 16: Differentiation 19 Exercise 15. Let µ be a complex measure on Rn . Let λ > 0 and K be a non-empty compact subset of {λ < M µ}. 1. Show that K can be covered by a finite collection Bi = B (xi , i ), i = 1, . . . , N of open balls, such that: ∀i = 1, . . . , N , λdx(Bi ) < |µ|(Bi ) 2. Show the existence of S ⊆ {1, . . . , N } such that: dx(K ) ≤ 3n λ−1 |µ| B (xi , i ) i∈S 3. Show that dx(K ) ≤ 3n λ−1 µ 4. Conclude with the following: Theorem 100 Let µ be a complex measure on (Rn , B (Rn )), n ≥ 1, with maximal function M µ. Then, for all λ ∈ R+ \ {0}, we have: dx({λ < M µ}) ≤ 3n λ−1 µ Tutorial 16: Differentiation 20 Definition 120 Let f ∈ L1 (Rn , B (Rn ), dx), and µ be the complex C measure µ = f dx on Rn , n ≥ 1. We call maximal function of f , denoted M f , the maximal function M µ of µ. Exercise 16. Let f ∈ L1 (Rn , B (Rn ), dx), n ≥ 1. C 1. Show that for all x ∈ Rn : (M f )(x) = sup >0 1 dx(B (x, )) |f |dx B (x, ) 2. Show that for all λ > 0, dx({λ < M f }) ≤ 3n λ−1 f 1. Definition 121 Let f ∈ L1 (Rn , B (Rn ), dx), n ≥ 1. We say that C x ∈ Rn is a lebesgue point of f , if and only if we have: lim ↓↓0 1 dx(B (x, )) |f (y ) − f (x)|dy = 0 B (x, ) Tutorial 16: Differentiation 21 Exercise 17. Let f ∈ L1 (Rn , B (Rn ), dx), n ≥ 1. C 1. Show that if f is continuous at x ∈ Rn , then x is a Lebesgue point of f . 2. Show that if x ∈ Rn is a Lebesgue point of f , then: f (x) = lim ↓↓0 1 dx(B (x, )) f (y )dy B (x, ) Exercise 18. Let n ≥ 1 and f ∈ L1 (Rn , B (Rn ), dx). For all C and x ∈ Rn , we define: (T f )(x) = 1 dx(B (x, )) |f (y ) − f (x)|dy B (x, ) and we put, for all x ∈ Rn : (T f )(x) = lim sup(T f )(x) = inf sup (Tu f )(x) ↓↓0 >0 u∈]0, [ >0 Tutorial 16: Differentiation 22 c 1. Given η > 0, show the existence of g ∈ CC (Rn ) such that: f −g 2. Let h = f − g . Show that for all (T h)(x) ≤ 1 dx(B (x, )) 1 ≤η > 0 and x ∈ Rn : |h|dx + |h(x)| B (x, ) 3. Show that T h ≤ M h + |h|. 4. Show that for all > 0, we have T f ≤ T g + T h. 5. Show that T f ≤ T g + T h. 6. Using the continuity of g , show that T g = 0. 7. Show that T f ≤ M h + |h|. 8. Show that for all α > 0, {2α < T f } ⊆ {α < M h} ∪ {α < |h|}. 9. Show that dx({α < |h|}) ≤ α−1 h 1 . Tutorial 16: Differentiation 23 10. Conclude that for all α > 0 and η > 0, there is Nα,η ∈ B (Rn ) such that {2α < T f } ⊆ Nα,η and dx(Nα,η ) ≤ η . 11. Show that for all α > 0, there exists Nα ∈ B (Rn ) such that {2α < T f } ⊆ Nα and dx(Nα ) = 0. 12. Show there is N ∈ B (Rn ), dx(N ) = 0, such that {T f > 0} ⊆ N . 13. Conclude that T f = 0 , dx−a.s. 14. Conclude with the following: Theorem 101 Let f ∈ L1 (Rn , B (Rn ), dx), n ≥ 1. Then, dx-almost C surely, any x ∈ Rn is a lebesgue points of f , i.e. 1 ↓↓0 dx(B (x, )) |f (y ) − f (x)|dy = 0 dx-a.s. , lim B (x, ) Tutorial 16: Differentiation 24 Exercise 19. Let (Ω, F , µ) be a measure space and Ω ∈ F . We define F = F|Ω and µ = µ|F . For all map f : Ω → [0, +∞] (or C), ˜ we define f : Ω → [0, +∞] (or C), by: ˜ f (ω ) = f (ω ) 0 ω∈Ω ω∈Ω if if 1. Show that F ⊆ F and conclude that µ is therefore a welldefined measure on (Ω , F ). 2. Let A ∈ F and 1A be the characteristic function of A defined on Ω . Let 1A be the characteristic function of A defined on Ω. Show that ˜A = 1A . 1 3. Let f : (Ω , F ) → [0, +∞] be a non-negative and measurable ˜ map. Show that f : (Ω, F ) → [0, +∞] is also non-negative and measurable, and that we have: ˜ f dµ f dµ = Ω Ω Tutorial 16: Differentiation 25 ˜ 4. Let f ∈ L1 (Ω , F , µ ). Show that f ∈ L1 (Ω, F , µ), and: C C ˜ f dµ f dµ = Ω Ω Definition 122 Let b : R+ → C be a right-continuous map of finite variation. We say that b is absolutely continuous, if and only if it is absolutely continuous with respect to a(t) = t. Exercise 20. Let b : R+ → C be right-continuous of finite variation. 1. Show that b is absolutely continuous, if and only if there is t f ∈ L1,loc (t) such that b(t) = 0 f (s)ds, for all t ∈ R+ . C 2. Show that b absolutely continuous ⇒ b continuous with b(0) = 0. Tutorial 16: Differentiation 26 Exercise 21. Let b : R+ → C be an absolutely continuous map. Let f ∈ L1,loc (t) be such that b = f.t. For all n ≥ 1, we define C fn : R → C by: fn (t) = f (t)1[0,n] (t) 0 if if t ∈ R+ t<0 1. Let n ≥ 1. Show fn ∈ L1 (R, B (R), dx) and for all t ∈ [0, n]: C t fn dx b(t) = 0 2. Show the existence of Nn ∈ B (R) such that dx(Nn ) = 0, and c for all t ∈ Nn , t is a Lebesgue point of fn . 3. Show that for all t ∈ R, and t+ 1 |fn (s) − fn (t)|ds ≤ t > 0: 2 dx(B (t, )) |fn (s) − fn (t)|ds B (t, ) Tutorial 16: Differentiation 27 c 4. Show that for all t ∈ Nn , we have: lim t+ 1 ↓↓0 fn (s)ds = fn (t) t c 5. Show similarly that for all t ∈ Nn , we have: lim ↓↓0 t 1 t− fn (s)ds = fn (t) c 6. Show that for all t ∈ Nn ∩ [0, n[, b (t) exists and b (t) = f (t).1 7. Show the existence of N ∈ B (R+ ), such that dx(N ) = 0, and: ∀t ∈ N c , b (t) exists with b (t) = f (t) 8. Conclude with the following: 1b (0) being a r.h.s derivative only. Tutorial 16: Differentiation 28 Theorem 102 A map b : R+ → C is absolutely continuous, if and only if there exists f ∈ L1,loc (t) such that: C t ∀t ∈ R+ , b(t) = f (s)ds 0 in which case, b is almost surely differentiable with b = f dx-a.s. Tutorial 17: Image Measure 1 17. Image Measure In the following, K denotes R or C. We denote Mn (K), n ≥ 1, the set of all n × n-matrices with K-valued entries. We recall that for all M = (mij ) ∈ Mn (K), M is identified with the linear map M : Kn → Kn uniquely determined by: n ∀j = 1, . . . , n , M ej = mij ei i=1 i n where (e1 , . . . , en ) is the canonical basis of K , i.e. ei = (0, ., 1 , ., 0). Exercise 1. For all α ∈ K, let Hα ∈ Mn (K) be defined by: α 10 Hα = .. . 0 1 Tutorial 17: Image Measure 2 i.e. by Hα e1 = αe1 , Hα ej = ej , for all j ≥ 2. For k, l ∈ {1, . . . , n}, we define the matrix Σkl ∈ Mn (K) by Σkl ek = el , Σkl el = ek and Σkl ej = ej , for all j ∈ {1, . . . , n} \ {k, l}. If n ≥ 2, we define the matrix U ∈ Mn (K) by: 10 1 1 0 U= .. . 0 1 i.e. by U e1 = e1 + e2 , U ej = ej for all j ≥ 2. If n = 1, we put U = 1. We define Nn (K) = {Hα : α ∈ K} ∪ {Σkl : k, l = 1, . . . , n} ∪ {U }, and Mn (K) to be the set of all finite products of elements of Nn (K): Mn (K) = {M ∈ Mn (K) : M = Q1 . . . . .Qp , p ≥ 1 , Qj ∈ Nn (K) , ∀j } We shall prove that Mn (K) = Mn (K). Tutorial 17: Image Measure 3 − 1. Show that if α ∈ K \ {0}, Hα is non-singular with Hα 1 = H1/α 2. Show that if k, l = 1, . . . , n, Σkl is non-singular with Σ−1 = Σkl . kl 3. Show that U is non-singular, and that for 10 −1 1 0 U −1 = . 0 .. n ≥ 2: 1 4. Let M = (mij ) ∈ Mn (K). Let R1 , . . . , Rn be the rows of M : R1 R2 M= . . . Rn Tutorial 17: Image Measure 4 Show that for all α ∈ K: Hα .M = αR1 R2 . . . Rn Conclude that multiplying M by Hα from the left, amounts to multiplying the first row of M by α. 5. Show that multiplying M by Hα from the right, amounts to multiplying the first column of M by α. 6. Show that multiplying M by Σkl from the left, amounts to swapping the rows Rl and Rk . 7. Show that multiplying M by Σkl from the right, amounts to swapping the columns Cl and Ck . Tutorial 17: Image Measure 5 8. Show that multiplying M by U −1 from the left ( n ≥ 2), amounts to subtracting R1 to R2 , i.e.: R1 R1 R2 R2 − R1 U −1 . . = . . . . . Rn Rn 9. Show that multiplying M by U −1 from the right (for n ≥ 2), amounts to subtracting C2 to C1 . 10. Define U = Σ12 .U −1 .Σ12 , (n ≥ 2). Show that multiplying M by U from the right, amounts to subtracting C1 to C2 . 11. Show that if n = 1, then indeed we have M1 (K) = M1 (K). Tutorial 17: Image Measure 6 Exercise 2. Further to exercise (1), we now assume that n ≥ 2, and make the induction hypothesis that Mn−1 (K) = Mn−1 (K). 1. Let On ∈ Mn (K) be the matrix with all entries equal to zero. Show the existence of Q1 , . . . , Qp ∈ Nn−1 (K), p ≥ 1, such that: On−1 = Q1 . . . . .Qp 2. For k = 1, . . . , p, we define Qk ∈ Mn (K), by: 0 . . Qk . Qk = 0 0 ... 0 1 Tutorial 17: Image Measure Show that Qk ∈ Nn (K), and that we have: 10 ... 0 Σ1n .Q1 . . . . .Qp .Σ1n = . . . On−1 0 7 0 3. Conclude that On ∈ Mn (K). 4. We now consider M = (mij ) ∈ Mn (K), M = On . We want to show that M ∈ Mn (K). Show that for some k, l ∈ {1, . . . , n}: 1 ∗ ... ∗ ∗ −1 Hmkl .Σ1k .M.Σ1l = . . . ∗ ∗ −1 5. Show that if Hmkl .Σ1k .M.Σ1l ∈ Mn (K), then M ∈ Mn (K). Conclude that without loss of generality, in order to prove that Tutorial 17: Image Measure 8 M lies in Mn (K) we can assume that m11 = 1. 6. Let i = 2, . . . , n. Show that if mi1 = 0, we have: 1 ∗ ... ∗ ∗ −1 −1 Hmi1 .Σ2i .U −1 .Σ2i .H1/mi1 .M = 0 ←i ∗ ∗ 7. Conclude that without loss of generality, we can assume that mi1 = 0 for all i ≥ 2, i.e. that M is of the form: 1 ∗ ... ∗ 0 M = . . . ∗ 0 8. Show that in order to prove that M ∈ Mn (K), without loss of Tutorial 17: Image Measure 9 generality, we can assume that M is of the form: 1 0 ... 0 0 M = . . . M 0 9. Prove that M ∈ Mn (K) and conclude with the following: Theorem 103 Given n ≥ 2, any n × n-matrix with values in K is a finite product of matrices Q of the following types: (i) (ii) (iii) Qe1 = αe1 , Qej = ej , ∀j = 2, . . . , n , (α ∈ K) Qel = ek , Qek = el , Qej = ej , ∀j = k, l , (k, l ∈ Nn ) Qe1 = e1 + e2 , Qej = ej , ∀j = 2, . . . , n where (e1 , . . . , en ) is the canonical basis of Kn . Tutorial 17: Image Measure 10 Definition 123 Let X : (Ω, F ) → (Ω , F ) be a measurable map, where (Ω, F ) and (Ω , F ) are two measurable spaces. Given a measure µ (possibly complex) on (Ω, F ), we call distribution of X under µ, or law of X under µ, or image measure of µ by X , the measure (possibly complex) denoted µX or X (µ) on (Ω , F ), defined by: ∀B ∈ F , µX (B ) = µ({X ∈ B }) = µ(X −1 (B )) Exercise 3. Let X : (Ω, F ) → (Ω , F ) be a measurable map, where (Ω, F ) and (Ω , F ) are two measurable spaces. 1. Show that if µ is a measure on (Ω, F ), µX is a well-defined measure on (Ω , F ). 2. Show that if µ is a complex measure on (Ω, F ), µX is a welldefined complex measure on (Ω , F ). 3. Let B ∈ F . Show that if (En )n≥1 is a measurable partition of B , then (X −1 (En ))n≥1 is a measurable partition of X −1 (B ). Tutorial 17: Image Measure 11 4. Show that if µ is a complex measure on (Ω, F ), then |µX | ≤ |µ|X . 5. Let Y : (Ω , F ) → (Ω , F ) be a measurable map, where (Ω , F ) is another measurable space. Show that for all (possibly complex) measure µ on (Ω, F ), we have: Y (X (µ)) = (Y ◦ X )(µ) = (µX )Y = µ(Y ◦X ) Definition 124 Let µ be a measure (possibly complex) on Rn , n ≥ 1. We say that µ is invariant by translation, if and only if for all a ∈ Rn , and associated translation mapping τa : Rn → Rn defined by τa (x) = a + x, we have τa (µ) = µ. Exercise 4. Let µ be a measure (possibly complex) on (Rn , B (Rn )). 1. Show that τa : (Rn , B (Rn )) → (Rn , B (Rn )) is measurable. 2. Show τa (µ) is therefore a well-defined (possibly complex) measure on (Rn , B (Rn )), for all a ∈ Rn . Tutorial 17: Image Measure 12 3. Show that τa (dx) = dx for all a ∈ Rn . 4. Show the lebesgue measure on Rn is invariant by translation. Exercise 5. Let kα : Rn → Rn be defined by kα (x) = αx, α > 0. 1. Show that kα : (Rn , B (Rn )) → (Rn , B (Rn )) is measurable. 2. Show that kα (dx) = α−n dx. Exercise 6. Show the following: Theorem 104 (Integral Projection 1) Let X : (Ω, F ) → (Ω , F ) be a measurable map, where (Ω, F ), (Ω , F ) are measurable spaces. Let µ be a measure on (Ω, F ). Then, for all f : (Ω , F ) → [0, +∞] non-negative and measurable, we have: f ◦ Xdµ = Ω f dX (µ) Ω Tutorial 17: Image Measure 13 Exercise 7. Show the following: Theorem 105 (Integral Projection 2) Let X : (Ω, F ) → (Ω , F ) be a measurable map, where (Ω, F ), (Ω , F ) are measurable spaces. Let µ be a measure on (Ω, F ). Then, for all f : (Ω , F ) → (C, B (C)) measurable, we have the equivalence: f ◦ X ∈ L1 (Ω, F , µ) ⇔ f ∈ L1 (Ω , F , X (µ)) C C in which case, we have: f ◦ Xdµ = Ω f dX (µ) Ω Exercise 8. Further to theorem (105), suppose µ is in fact a complex measure on (Ω, F ). Show that: |f |d|X (µ)| ≤ Ω Conclude with the following: |f ◦ X |d|µ| Ω (1) Tutorial 17: Image Measure 14 Theorem 106 (Integral Projection 3) Let X : (Ω, F ) → (Ω , F ) be a measurable map, where (Ω, F ), (Ω , F ) are measurable spaces. Let µ be a complex measure on (Ω, F ). Then, for all measurable map f : (Ω , F ) → (C, B (C)), we have: f ◦ X ∈ L1 (Ω, F , µ) ⇒ f ∈ L1 (Ω , F , X (µ)) C C and when the left-hand side of this implication is satisfied: f ◦ Xdµ = Ω f dX (µ) Ω Exercise 9. Let X : (Ω, F ) → (Rn , B (Rn )) be a measurable map with distribution µ = X (P ), where (Ω, F , P ) is a probability space. 1. Show that X is integrable, if and only if: +∞ −∞ |x|dµ(x) < +∞ Tutorial 17: Image Measure 15 2. Show that if X is integrable, then: +∞ xdµ(x) E [X ] = −∞ 3. Show that: E [X 2 ] = +∞ x2 dµ(x) −∞ Exercise 10. Let µ be a locally finite measure on (Rn , B (Rn )), which is invariant by translation. For all a = (a1 , . . . , an ) ∈ (R+ )n , we define Qa = [0, a1 [× . . . × [0, an [, and in particular Q = Q(1,...,1) = [0, 1[n . 1. Show that µ(Qa ) < +∞ for all a ∈ (R+ )n , and µ(Q) < +∞. 2. Let p = (p1 , . . . , pn ) where pi ≥ 1 is an integer for all i’s. Show: Qp = [k1 , k1 + 1[× . . . × [kn , kn + 1[ k = (k1 , . . . , kn ) 0 ≤ ki < pi Tutorial 17: Image Measure 16 3. Show that µ(Qp ) = p1 . . . pn µ(Q). 4. Let q1 , . . . , qn ≥ 1 be n positive integers. Show that: k1 p1 (k1 + 1)p1 kn pn (kn + 1)pn , [× . . . × [ , [ q1 q1 qn qn k = (k1 , . . . , kn ) 0 ≤ ki < qi Qp = [ 5. Show that µ(Qp ) = q1 . . . qn µ(Q(p1 /q1 ,...,pn /qn ) ) 6. Show that µ(Qr ) = r1 . . . rn µ(Q), for all r ∈ (Q+ )n . 7. Show that µ(Qa ) = a1 . . . an µ(Q), for all a ∈ (R+ )n . 8. Show that µ(B ) = µ(Q)dx(B ), for all B ∈ C , where: C = {[a1 , b1 [× . . . × [an , bn [ , ai , bi ∈ R , ai ≤ bi , ∀i ∈ Nn } 9. Show that B (Rn ) = σ (C ). Tutorial 17: Image Measure 17 10. Show that µ = µ(Q)dx, and conclude with the following: Theorem 107 Let µ be a locally finite measure on (Rn , B (Rn )). If µ is invariant by translation, then there exists α ∈ R+ such that: µ = αdx Exercise 11. Let T : Rn → Rn be a linear bijection. 1. Show that T and T −1 are continuous. 2. Show that for all B ⊆ Rn , the inverse image T −1 (B ) = {T ∈ B } coincides with the direct image: T −1 (B ) = {y : y = T −1 (x) for some x ∈ B } 3. Show that for all B ⊆ Rn , the direct image T (B ) coincides with the inverse image (T −1 )−1 (B ) = {T −1 ∈ B }. 4. Let K ⊆ Rn be compact. Show that {T ∈ K } is compact. Tutorial 17: Image Measure 18 5. Show that T (dx) is a locally finite measure on (Rn , B (Rn )). 6. Let τa be the translation of vector a ∈ Rn . Show that: T ◦ τT −1 (a) = τa ◦ T 7. Show that T (dx) is invariant by translation. 8. Show the existence of α ∈ R+ , such that T (dx) = αdx. Show that such constant is unique, and denote it by ∆(T ). 9. Show that Q = T ([0, 1]n) ∈ B (Rn ) and that we have: ∆(T )dx(Q) = T (dx)(Q) = 1 10. Show that ∆(T ) = 0. 11. Let T1 , T2 : Rn → Rn be two linear bijections. Show that: (T1 ◦ T2 )(dx) = ∆(T1 )∆(T2 )dx and conclude that ∆(T1 ◦ T2 ) = ∆(T1 )∆(T2 ). Tutorial 17: Image Measure 19 Exercise 12. Let α ∈ R \ {0}. Let Hα : Rn → Rn be the linear bijection uniquely defined by Hα (e1 ) = αe1 , Hα (ej ) = ej for j ≥ 2. 1. Show that Hα (dx)([0, 1]n ) = |α|−1 . 2. Conclude that ∆(Hα ) = | det Hα |−1 . Exercise 13. Let k, l ∈ Nn and Σ : Rn → Rn be the linear bijection uniquely defined by Σ(ek ) = el , Σ(el ) = ek , Σ(ej ) = ej , for j = k, l. 1. Show that Σ(dx)([0, 1]n ) = 1. 2. Show that Σ.Σ = In . (Identity mapping on Rn ). 3. Show that | det Σ| = 1. 4. Conclude that ∆(Σ) = | det Σ|−1 . Exercise 14. Let n ≥ 2 and U : Rn → Rn be the linear bijection uniquely defined by U (e1 ) = e1 + e2 and U (ej ) = ej for j ≥ 2. Let Q = [0, 1[n . Tutorial 17: Image Measure 20 1. Show that: U −1 (Q) = {x ∈ Rn : 0 ≤ x1 + x2 < 1 , 0 ≤ xi < 1 , ∀i = 2} 2. Define: Ω1 = U −1 (Q) ∩ {x ∈ Rn : x2 ≥ 0} Ω2 = U −1 (Q) ∩ {x ∈ Rn : x2 < 0} Show that Ω1 , Ω2 ∈ B (Rn ). 3. Let τe2 be the translation of vector e2 . Draw a picture of Q, Ω1 , Ω2 and τe2 (Ω2 ) in the case when n = 2. 4. Show that if x ∈ Ω1 , then 0 ≤ x2 < 1. 5. Show that Ω1 ⊆ Q. 6. Show that if x ∈ τe2 (Ω2 ), then 0 ≤ x2 < 1. 7. Show that τe2 (Ω2 ) ⊆ Q. Tutorial 17: Image Measure 21 8. Show that if x ∈ Q and x1 + x2 < 1 then x ∈ Ω1 . 9. Show that if x ∈ Q and x1 + x2 ≥ 1 then x ∈ τe2 (Ω2 ). 10. Show that if x ∈ τe2 (Ω2 ) then x1 + x2 ≥ 1. 11. Show that τe2 (Ω2 ) ∩ Ω1 = ∅. 12. Show that Q = Ω1 τe2 (Ω2 ). 13. Show that dx(Q) = dx(U −1 (Q)). 14. Show that ∆(U ) = 1. 15. Show that ∆(U ) = | det U |−1 . Exercise 15. Let T : Rn → Rn be a linear bijection, (n ≥ 1). 1. Show the existence of linear bijections Q1 , . . . , Qp : Rn → Rn , p ≥ 1, with T = Q1 ◦ . . . ◦ Qp , ∆(Qi ) = | det Qi |−1 for all i ∈ Np . Tutorial 17: Image Measure 22 2. Show that ∆(T ) = | det T |−1 . 3. Conclude with the following: Theorem 108 Let n ≥ 1 and T : Rn → Rn be a linear bijection. Then, the image measure T (dx) of the lebesgue measure on Rn is: T (dx) = | det T |−1 dx Exercise 16. Let f : (R2 , B (R2 )) → [0, +∞] be a non-negative and measurable map. Let a, b, c, d ∈ R such that ad − bc = 0. Show that: f (ax + by, cx + dy )dxdy = |ad − bc|−1 R2 f (x, y )dxdy R2 Exercise 17. Let T : Rn → Rn be a linear bijection. Show that for all B ∈ B (Rn ), we have T (B ) ∈ B (Rn ) and: dx(T (B )) = | det T |dx(B ) Tutorial 17: Image Measure 23 Exercise 18. Let V be a linear subspace of Rn and p = dim V . We assume that 1 ≤ p ≤ n − 1. Let u1 , . . . , up be an orthonormal basis of V , and up+1 , . . . , un be such that u1 , . . . , un is an orthonormal basis of Rn . For i ∈ Nn , Let φi : Rn → R be defined by φi (x) = ui , x . 1. Show that all φi ’s are continuous. 2. Show that V = n j =p+1 φ−1 ({0}). j 3. Show that V is a closed subset of Rn . 4. Let Q = (qij ) ∈ Mn (R) be the matrix uniquely defined by Qej = uj for all j ∈ Nn , where (e1 , . . . , en ) is the canonical basis of Rn . Show that for all i, j ∈ Nn : n ui , uj = qki qkj k=1 5. Show that QT .Q = Q.QT = In and conclude that | det Q| = 1. Tutorial 17: Image Measure 24 6. Show that dx({Q ∈ V }) = dx(V ). 7. Show that {Q ∈ V } = span(e1 , . . . , ep ).1 8. For all m ≥ 1, we define: n−1 Em = [−m, m] × . . . × [−m, m] ×{0} Show that dx(Em ) = 0 for all m ≥ 1. 9. Show that dx(span(e1 , . . . , en−1 )) = 0. 10. Conclude with the following: Theorem 109 Let n ≥ 1. Any linear subspace V of Rn is a closed subset of Rn . Moreover, if dim V ≤ n − 1, then dx(V ) = 0. 1 i.e. the linear subspace of Rn generated by e1 , . . . , ep . Tutorial 18: The Jacobian Formula 1 18. The Jacobian Formula In the following, K denotes R or C. Definition 125 We call K-normed space, an ordered pair (E, N ), where E is a K-vector space, and N : E → R+ is a norm on E . See definition (89) for vector space, and definition (95) for norm. Exercise 1. Let (H, ·, · ) be a K-hilbert space, and 1. Show that · ·= ·, · . is a norm on H. 2. Show that (H, · ) is a K-normed space. Exercise 2. Let (E, · ) be a K-normed space: 1. Show that d(x, y ) = x − y defines a metric on E . 2. Show that for all x, y ∈ E , we have | x − y | ≤ x − y . Tutorial 18: The Jacobian Formula 2 Definition 126 Let (E, · ) be a K-normed space, and d be the metric defined by d(x, y ) = x − y . We call norm topology on E , denoted T · , the topology on E associated with d. Exercise 3. Let E, F be two K-normed spaces, and l : E → F be a linear map. Show that the following are equivalent: (i) (ii) (iii) (iv ) l is continuous (w.r. to the norm topologies) l is continuous at x = 0. ∃K ∈ R+ , ∀x ∈ E , l(x) ≤ K x sup{ l(x) : x ∈ E , x = 1} < +∞ Definition 127 Let E , F be K-normed spaces. The K-vector space of all continuous linear maps l : E → F is denoted LK (E, F ). Exercise 4. Show that LK (E, F ) is indeed a K-vector space. Tutorial 18: The Jacobian Formula 3 Exercise 5. Let E, F be K-normed spaces. Given l ∈ LK (E, F ), let: l = sup{ l(x) : x ∈ E , x = 1} < +∞ 1. Show that: l = sup{ l(x) : x ∈ E , x ≤ 1} 2. Show that: l = sup l(x) : x∈E , x=0 x 3. Show that l(x) ≤ l . x , for all x ∈ E . 4. Show that l is the smallest K ∈ R+ , such that: ∀x ∈ E , l(x) ≤ K x 5. Show that l → l is a norm on LK (E, F ). 6. Show that (LK (E, F ), · ) is a K-normed space. Tutorial 18: The Jacobian Formula 4 Definition 128 Let E, F be R-normed spaces and U be an open subset of E . We say that a map φ : U → F is differentiable at some a ∈ U , if and only if there exists l ∈ LR (E, F ) such that, for all > 0, there exists δ > 0, such that for all h ∈ E : h ≤ δ ⇒ a + h ∈ U and φ(a + h) − φ(a) − l(h) ≤ h Exercise 6. Let E, F be two R-normed spaces, and U be open in E . Let φ : U → F be a map and a ∈ U . 1. Suppose that φ : U → F is differentiable at a ∈ U , and that l1 , l2 ∈ LR (E, F ) satisfy the requirement of definition (128). Show that for all > 0, there exists δ > 0 such that: ∀h ∈ E , h ≤ δ ⇒ l1 (h) − l2 (h) ≤ h 2. Conclude that l1 − l2 = 0 and finally that l1 = l2 . Tutorial 18: The Jacobian Formula 5 Definition 129 Let E, F be R-normed spaces and U be an open subset of E . Let φ : U → F be a map and a ∈ U . If φ is differentiable at a, we call differential of φ at a, the unique element of LR (E, F ), denoted dφ(a), satisfying the requirement of definition (128). If φ is differentiable at all points of U , the map dφ : U → LR (E, F ) is also called the differential of φ. Definition 130 Let E, F be R-normed spaces and U be an open subset of E . A map φ : U → F is said to be of class C 1 , if and only if dφ(a) exists for all a ∈ U , and the differential dφ : U → LR (E, F ) is a continuous map. Exercise 7. Let E, F be two R-normed spaces and U be open in E . Let φ : U → F be a map, and a ∈ U . 1. Show that φ differentiable at a ⇒ φ continuous at a. 2. If φ is of class C 1 , explain with respect to which topologies the differential dφ : U → LR (E, F ) is said to be continuous. Tutorial 18: The Jacobian Formula 6 3. Show that if φ is of class C 1 , then φ is continuous. 4. Suppose that E = R. Show that for all a ∈ U , φ is differentiable at a ∈ U , if and only if the derivative: φ (a) = lim t=0,t→0 φ(a + t) − φ(a) t exists in F , in which case dφ(a) ∈ LR (R, F ) is given by: ∀t ∈ R , dφ(a)(t) = t.φ (a) In particular, φ (a) = dφ(a)(1). Exercise 8. Let E, F, G be three R-normed spaces. Let U be open in E and V be open in F . Let φ : U → F and ψ : V → G be two maps such that φ(U ) ⊆ V . We assume that φ is differentiable at a ∈ U , and we put: l1 = dφ(a) ∈ LR (E, F ) Tutorial 18: The Jacobian Formula 7 We assume that ψ is differentiable at φ(a) ∈ V , and we put: l2 = dψ (φ(a)) ∈ LR (F, G) 1. Explain why ψ ◦ φ : U → G is a well-defined map. 2. Given > 0, show the existence of η > 0 such that: η (η + l1 + l2 ) ≤ 3. Show the existence of δ2 > 0 such that for all h2 ∈ F with h2 ≤ δ2 , we have φ(a) + h2 ∈ V and: ψ (φ(a) + h2 ) ψ ◦ φ(a) − l2 (h2 ) ≤ η h2 4. Show that if h2 ∈ F and h2 ≤ δ2 , then for all h ∈ E , we have: ψ (φ(a) + h2 ) − ψ ◦ φ(a) − l2 ◦ l1 (h) ≤ η h2 + l2 . h2 − l1 (h) 5. Show the existence of δ > 0 such that for all h ∈ E with h ≤ δ , we have a + h ∈ U and φ(a + h) − φ(a) − l1(h) ≤ η h , together with φ(a + h) − φ(a) ≤ δ2 . Tutorial 18: The Jacobian Formula 8 6. Show that if h ∈ E is such that h ≤ δ , then a + h ∈ U and: ψ ◦ φ(a+h) − ψ ◦ φ(a) − l2 ◦ l1 (h) ≤ η φ(a+h)−φ(a) +η l2 . h ≤ η (η + l1 + l2 ) h ≤h 7. Show that l2 ◦ l1 ∈ LR (E, G) 8. Conclude with the following: Theorem 110 Let E, F, G be three R-normed spaces, U be open in E and V be open in F . Let φ : U → F and ψ : V → G be two maps such that φ(U ) ⊆ V . Let a ∈ U . Then, if φ is differentiable at a ∈ U , and ψ is differentiable at φ(a) ∈ V , then ψ ◦ φ is differentiable at a ∈ U , and furthermore: d(ψ ◦ φ)(a) = dψ (φ(a)) ◦ dφ(a) Tutorial 18: The Jacobian Formula 9 Exercise 9. Let E, F, G be three R-normed spaces. Let U be open in E and V be open in F . Let φ : U → F and ψ : V → G be two maps of class C 1 such that φ(U ) ⊆ V . 1. For all (l1 , l2 ) ∈ LR (F, G) × LR (E, F ), we define: N1 (l1 , l2 ) = N2 (l1 , l2 ) = N∞ (l1 , l2 ) = l1 + l2 l1 2 + l2 2 max( l1 , l2 ) Show that N1 , N2 , N∞ are all norms on LR (F, G) × LR (E, F ). 2. Show they induce the product topology on LR (F, G)×LR (E, F ). 3. We define the map H : LR (F, G) × LR (E, F ) → LR (E, G) by: ∀(l1 , l2 ) ∈ LR (F, G) × LR (E, F ) , H (l1 , l2 ) = l1 ◦ l2 Show that H (l1 , l2 ) ≤ l1 . l2 , for all l1 , l2 . Tutorial 18: The Jacobian Formula 10 4. Show that H is continuous. 5. We define K : U → LR (F, G) × LR (E, F ) by: ∀a ∈ U , K (a) = (dψ (φ(a)), dφ(a)) Show that K is continuous. 6. Show that ψ ◦ φ is differentiable on U . 7. Show that d(ψ ◦ φ) = H ◦ K . 8. Conclude with the following: Theorem 111 Let E, F, G be three R-normed spaces, U be open in E and V be open in F . Let φ : U → F and ψ : V → G be two maps of class C 1 such that φ(U ) ⊆ V . Then, ψ ◦ φ : U → G is of class C 1 . Tutorial 18: The Jacobian Formula 11 Exercise 10. Let E be an R-normed space. Let a, b ∈ R, a < b. Let f : [a, b] → E and g : [a, b] → R be two continuous maps which are differentiable at every points of ]a, b[. We assume that: ∀t ∈]a, b[ , 1. Given f (t) ≤ g (t) > 0, we define φ : [a, b] → R by: φ (t) = f (t) − f (a) − g (t) + g (a) − (t − a) for all t ∈ [a, b]. Show that φ is continuous. 2. Define E = {t ∈ [a, b] : φ (t) ≤ }, and c = sup E . Show that: c ∈ [a, b] and φ (c) ≤ 3. Show the existence of h > 0, such that: ∀t ∈ [a, a + h[∩[a, b] , φ (t) ≤ 4. Show that c ∈]a, b]. Tutorial 18: The Jacobian Formula 12 5. Suppose that c ∈]a, b[. Show the existence of t0 ∈]c, b] such that: g (t0 ) − g (c) f (t0 ) − f (c) ≤ f (c) + /2 and g (c) ≤ + /2 t0 − c t0 − c 6. Show that f (t0 ) − f (c) ≤ g (t0 ) − g (c) + (t0 − c). 7. Show that f (c) − f (a) ≤ g (c) − g (a) + (c − a) + . 8. Show that f (t0 ) − f (a) ≤ g (t0 ) − g (a) + (t0 − a) + . 9. Show that c cannot be the supremum of E unless c = b. 10. Show that f (b) − f (a) ≤ g (b) − g (a) + (b − a) + . 11. Conclude with the following: Tutorial 18: The Jacobian Formula 13 Theorem 112 Let E be an R-normed space. Let a, b ∈ R, a < b. Let f : [a, b] → E and g : [a, b] → R be two continuous maps which are differentiable at every point of ]a, b[, and such that: ∀t ∈]a, b[ , f (t) ≤ g (t) Then: f (b) − f (a) ≤ g (b) − g (a) Definition 131 Let n ≥ 1 and U be open in Rn . Let φ : U → E be a map, where E is an R-normed space. For all i = 1, . . . , n, we say that φ has an ith partial derivative at a ∈ U , if and only if the limit: φ(a + hei ) − φ(a) ∂φ (a) = lim h=0,h→0 ∂xi h exists in E , where (e1 , . . . , en ) is the canonical basis of Rn . Tutorial 18: The Jacobian Formula 14 Exercise 11. Let n ≥ 1 and U be open in Rn . Let φ : U → E be a map, where E is an R-normed space. 1. Suppose φ is differentiable at a ∈ U . Show that for all i ∈ Nn : 1 φ(a + hei ) − φ(a) − dφ(a)(hei ) = 0 lim h=0,h→0 hei 2. Show that for all i ∈ Nn , ∂φ ∂xi (a) exists, and: ∂φ (a) = dφ(a)(ei ) ∂xi 3. Conclude with the following: Theorem 113 Let n ≥ 1 and U be open in Rn . Let φ : U → E be a map, where E is an R-normed space. Then, if φ is differentiable at ∂φ a ∈ U , for all i = 1, . . . , n, ∂xi (a) exists and we have: n ∀h = (h1 , . . . , hn ) ∈ R , dφ(a)(h) = n i=1 ∂φ (a)hi ∂xi Tutorial 18: The Jacobian Formula 15 Exercise 12. Let n ≥ 1 and U be open in Rn . Let φ : U → E be a map, where E is an R-normed space. 1. Show that if φ is differentiable at a, b ∈ U , then for all i ∈ Nn : ∂φ ∂φ (b) − (a) ≤ dφ(b) − dφ(a) ∂xi ∂xi 2. Conclude that if φ is of class C 1 on U , then continuous on U , for all i ∈ Nn . ∂φ ∂xi exists and is Exercise 13. Let n ≥ 1 and U be open in Rn . Let φ : U → E be a ∂φ map, where E is an R-normed space. We assume that ∂xi exists on U , and is continuous at a ∈ U , for all i ∈ Nn . We define l : Rn → E by: n ∂φ (a)hi ∀h = (h1 , . . . , hn ) ∈ Rn , l(h) = ∂xi i=1 1. Show that l ∈ LR (Rn , E ). Tutorial 18: The Jacobian Formula 16 2. Given > 0, show the existence of η > 0 such that for all h ∈ Rn with h < η , we have a + h ∈ U , and: ∀i = 1, . . . , n , ∂φ ∂φ (a + h) − (a) ≤ ∂xi ∂xi 3. Let h = (h1 , . . . , hn ) ∈ Rn be such that h < η . (e1 , . . . , en ) being the canonical basis of Rn , we define k0 = a and for i ∈ Nn : i ki = a + hi e i j =1 Show that k0 , . . . , kn ∈ U , and that we have: n φ(ki−1 + hi ei ) − φ(ki−1 ) − hi φ(a+h)−φ(a)−l(h) = i=1 ∂φ (a) ∂xi 4. Let i ∈ Nn . Assume that hi > 0. We define fi : [0, hi ] → E by: ∀t ∈ [0, hi ] , fi (t) = φ(ki−1 + tei ) − φ(ki−1 ) − t ∂φ (a) ∂xi Tutorial 18: The Jacobian Formula 17 Show fi is well-defined, fi (t) exists for all t ∈ [0, hi ], and: ∀t ∈ [0, hi ] , fi (t) = ∂φ ∂φ (ki−1 + tei ) − (a) ∂xi ∂xi 5. Show fi is continuous on [0, hi ], differentiable on ]0, hi [, with: ∀t ∈]0, hi [ , fi (t) ≤ 6. Show that: φ(ki−1 + hi ei ) − φ(ki−1 ) − hi ∂φ (a) ≤ |hi | ∂xi 7. Show that the previous inequality still holds if hi ≤ 0. 8. Conclude that for all h ∈ Rn with h < η , we have: √ nh φ(a + h) − φ(a) − l(h) ≤ 9. Prove the following: Tutorial 18: The Jacobian Formula 18 Theorem 114 Let n ≥ 1 and U be open in Rn . Let φ : U → E be ∂φ a map, where E is an R-normed space. If, for all i ∈ Nn ∂xi exists on U and is continuous at a ∈ U , then φ is differentiable at a ∈ U . Exercise 14. Let n ≥ 1 and U be open in Rn . Let φ : U → E be a map, where E is an R-normed space. We assume that for all i ∈ Nn , ∂φ ∂xi exists and is continuous on U . 1. Show that φ is differentiable on U . 2. Show that for all a, b ∈ U and h ∈ Rn : n (dφ(b) − dφ(a))(h) ≤ i=1 ∂φ ∂φ (b) − (a) ∂xi ∂xi 2 ∂φ ∂φ (b) − (a) ∂xi ∂xi 2 1/2 h 3. Show that for all a, b ∈ U : n dφ(b) − dφ(a) ≤ i=1 1/2 Tutorial 18: The Jacobian Formula 19 4. Show that dφ : U → LR (Rn , E ) is continuous. 5. Prove the following: Theorem 115 Let n ≥ 1 and U be open in Rn . Let φ : U → E be a map, where E is an R-normed space. Then, φ is of class C 1 on U , ∂φ if and only if for all i = 1, . . . , n, ∂xi exists and is continuous on U . Exercise 15. Let E, F be two R-normed spaces and l ∈ LR (E, F ). Let U be open in E and l|U be the restriction of l to U . Show that l|U is of class C 1 on U , and that we have: ∀x ∈ U , d(l|U )(x) = l Tutorial 18: The Jacobian Formula 20 Exercise 16. Let E1 , . . . , Ep , (p ≥ 1), be p R-normed spaces. Let E = E1 × . . . Ep . For all x = (x1 , . . . , xp ) ∈ E , we define: p x 1 = xi i=1 p x 2 = xi 2 i=1 x 1. Show that . 1 , . 2. Show . 1 , . 2 ∞ 2 = and . and . ∞ max i=1,...,p ∞ xi are all norms on E . induce the product topology on E . 3. Conclude that E is also an R-normed space, and that the norm topology on E is exactly the product topology on E . Tutorial 18: The Jacobian Formula 21 Exercise 17. Let E, F1 , . . . , Fp , (p ≥ 1) be p + 1 R-normed spaces, U be open in E , F = F1 × . . . × Fp and φ : U → F be a map. 1. For i = 1, . . . , p, let pi : F → Fi be the canonical projection. Show that pi ∈ LR (F, Fi ). We put φi = pi ◦ φ. 2. For i = 1, . . . , p, let ui : Fi → F be defined by: i ∀xi ∈ Fi , ui (xi ) = (0, . . . , xi , . . . , 0) Show that ui ∈ LR (Fi , F ) and φ = p i=1 ui ◦ φi . 3. Show that if φ is differentiable at a ∈ U , then for all i = 1, . . . , p, φi : U → Fi is differentiable at a ∈ U and dφi (a) = pi ◦ dφ(a). 4. Show that if for all i = 1, . . . , p, φi is differentiable at a ∈ U , then φ is differentiable at a ∈ U and: p ui ◦ dφi (a) dφ(a) = i=1 Tutorial 18: The Jacobian Formula 22 5. Suppose that φ is differentiable at a, b ∈ U . We assume that F p 2 is given the norm (x1 , . . . , xp ) 2 = i=1 xi . Show that for all i ∈ Np : dφi (b) − dφi (a) ≤ dφ(b) − dφ(a) 6. Show that: p dφi (b) − dφi (a) dφ(b) − dφ(a) ≤ 2 i=1 7. Show that φ is of class C 1 ⇔ φi is of class C 1 for all i ∈ Np . 8. Explain why this conclusion would still hold, if F were given the norm . 1 or . ∞ instead of . 2 . 9. Conclude with theorem (116) Tutorial 18: The Jacobian Formula 23 Theorem 116 Let E, F1 , . . . , Fp , (p ≥ 1), be p + 1 R-normed spaces and U be open in E . Let F be the R-normed space F = F1 × . . . × Fp and φ = (φ1 , . . . , φp ) : U → F be a map. Then, φ is differentiable at a ∈ U , if and only if dφi (a) exists for all i ∈ Np , in which case: ∀h ∈ E , dφ(a)(h) = (dφ1 (a)(h), . . . , dφp (a)(h)) Also, φ is of class C 1 on U ⇔ φi is of class C 1 on U , for all i ∈ Np . Theorem 117 Let φ = (φ1 , . . . , φn ) : U → Rn be a map, where n ≥ 1 and U is open in Rn . We assume that φ is differentiable at ∂φi a ∈ U . Then, for all i, j = 1, . . . , n, ∂xj (a) exists, and we have: ∂φ1 ∂φ1 ∂x1 (a) . . . ∂xn (a) . . . . dφ(a) = . . ∂φn ∂x1 (a) ... ∂φn ∂xn (a) Moreover, φ is of class C 1 on U , if and only if for all i, j = 1, . . . , n, ∂φi ∂xj exists and is continuous on U . Tutorial 18: The Jacobian Formula 24 Exercise 18. Prove theorem (117) Definition 132 Let φ = (φ1 , . . . , φn ) : U → Rn be a map, where n ≥ 1 and U is open in Rn . We assume that φ is differentiable at a ∈ U . We call jacobian of φ at a, denoted J (φ)(a), the determinant of the differential dφ(a) at a, i.e. ∂φ1 ∂φ1 . . . ∂xn (a) ∂x1 (a) . . . . J (φ)(a) = det . . ∂φn ∂φn ∂x1 (a) . . . ∂xn (a) Definition 133 Let n ≥ 1 and Ω, Ω be open in Rn . A bijection φ : Ω → Ω is called a C 1 -diffeomorphism between Ω and Ω , if and only if φ : Ω → Rn and φ−1 : Ω → Rn are both of class C 1 . Tutorial 18: The Jacobian Formula 25 Exercise 19. Let Ω and Ω be open in Rn . Let φ : Ω → Ω be a C 1 -diffeomorphism, ψ = φ−1 , and In be the identity mapping of Rn . 1. Explain why J (ψ ) : Ω → R and J (φ) : Ω → R are continuous. 2. Show that dφ(ψ (x)) ◦ dψ (x) = In , for all x ∈ Ω . 3. Show that dψ (φ(x)) ◦ dφ(x) = In , for all x ∈ Ω. 4. Show that J (ψ )(x) = 0 for all x ∈ Ω . 5. Show that J (φ)(x) = 0 for all x ∈ Ω. 6. Show that J (ψ ) = 1/(J (φ) ◦ ψ ) and J (φ) = 1/(J (ψ ) ◦ φ). Definition 134 Let n ≥ 1 and Ω ∈ B (Rn ), be a borel set in Rn . We define the lebesgue measure on Ω, denoted dx|Ω , as the restriction to B (Ω) of the lebesgue measure on Rn , i.e the measure on (Ω, B (Ω)) defined by: ∀B ∈ B (Ω) , dx|Ω (B ) = dx(B ) Tutorial 18: The Jacobian Formula 26 Exercise 20. Show that dx|Ω is a well-defined measure on (Ω, B (Ω)). Exercise 21. Let n ≥ 1 and Ω, Ω be open in Rn . Let φ : Ω → Ω be a C 1 -diffeomorphism and ψ = φ−1 . Let a ∈ Ω . We assume that dψ (a) = In , (identity mapping on Rn ), and given > 0, we denote: B (a, ) = {x ∈ Rn : a−x < } n where . is the usual norm in R . 1. Why are dx|Ω , φ(dx|Ω ) well-defined measures on (Ω , B (Ω )). 2. Show that for > 0 sufficiently small, B (a, ) ∈ B (Ω ). 3. Show that it makes sense to investigate whether the limit: φ(dx|Ω )(B (a, )) ↓↓0 dx|Ω (B (a, )) lim does exists in R. Tutorial 18: The Jacobian Formula 27 4. Given r > 0, show the existence of 1 > 0 such that for all h ∈ Rn with h ≤ 1 , we have a + h ∈ Ω , and: ψ (a + h) − ψ (a) − h ≤ r h 5. Show for all h ∈ Rn with h ≤ 1, we have a + h ∈ Ω , and: ψ (a + h) − ψ (a) ≤ (1 + r) h 6. Show that for all ∈]0, 1 [, we have B (a, ) ⊆ Ω , and: ψ (B (a, )) ⊆ B (ψ (a), (1 + r)) 7. Show that dφ(ψ (a)) = In . 8. Show the existence of 2 > 0 such that for all k ∈ Rn with k ≤ 2 , we have ψ (a) + k ∈ Ω, and: φ(ψ (a) + k ) − a − k ≤ r k Tutorial 18: The Jacobian Formula 28 9. Show for all k ∈ Rn with k ≤ 2, we have ψ (a) + k ∈ Ω, and: φ(ψ (a) + k ) − a ≤ (1 + r) k ∈]0, 10. Show for all 2 (1 + r)[, we have B (ψ (a), 1+r ) ⊆ Ω, and: B (ψ (a), 1+r ) ⊆ {φ ∈ B (a, )} 11. Show that if B (a, ) ⊆ Ω , then ψ (B (a, )) = {φ ∈ B (a, )}. 12. Show if 0 < < B (ψ (a), (ii) (iii) = 1+r 13. Show that for all (i) 0 1 ∧ 2 (1 + r), then B (a, ) ⊆ Ω , and: ) ⊆ {φ ∈ B (a, )} ⊆ B (ψ (a), (1 + r)) ∈]0, 0 [: )) = (1 + r)−n dx|Ω (B (a, )) 1+r dx(B (ψ (a), (1 + r))) = (1 + r)n dx|Ω (B (a, )) dx({φ ∈ B (a, )}) = φ(dx|Ω )(B (a, )) dx(B (ψ (a), Tutorial 18: The Jacobian Formula 14. Show that for all ∈]0, (1 + r)−n ≤ 0 [, 29 B (a, ) ⊆ Ω , and: φ(dx|Ω )(B (a, )) ≤ (1 + r)n dx|Ω (B (a, )) 15. Conclude that: lim ↓↓0 φ(dx|Ω )(B (a, )) =1 dx|Ω (B (a, )) Exercise 22. Let n ≥ 1 and Ω, Ω be open in Rn . Let φ : Ω → Ω be a C 1 -diffeomorphism and ψ = φ−1 . Let a ∈ Ω . We put A = dψ (a). 1. Show that A : Rn → Rn is a linear bijection. 2. Define Ω = A−1 (Ω). Show that this definition does not depend on whether A−1 (Ω) is viewed as inverse , or direct image. 3. Show that Ω is an open subset of Rn . Tutorial 18: The Jacobian Formula 30 ˜ ˜ ˜ 4. We define φ : Ω → Ω by φ(x) = φ ◦ A(x). Show that φ is a 1 ˜ = φ−1 = A−1 ◦ ψ . ˜ C -diffeomorphism with ψ ˜ 5. Show that dψ (a) = In . 6. Show that: ˜ φ(dx|Ω )(B (a, )) =1 ↓↓0 dx|Ω (B (a, )) lim 7. Let > 0 with B (a, ) ⊆ Ω . Justify each of the following steps: ˜ ˜ φ(dx|Ω )(B (a, )) = dx|Ω ({φ ∈ B (a, )}) ˜ = dx({φ ∈ B (a, )}) (1) (2) = dx({x ∈ Ω : φ ◦ A(x) ∈ B (a, )}) (3) = dx({x ∈ Ω : A(x) ∈ φ−1 (B (a, ))}) (4) = dx({x ∈ Rn : A(x) ∈ φ−1 (B (a, ))})(5) = A(dx)({φ ∈ B (a, )}) (6) = | det A|−1 dx({φ ∈ B (a, )}) (7) Tutorial 18: The Jacobian Formula 31 = | det A|−1 dx|Ω ({φ ∈ B (a, )}) −1 = | det A| 8. Show that: lim ↓↓0 φ(dx|Ω )(B (a, )) (8) (9) φ(dx|Ω )(B (a, )) = | det A| dx|Ω (B (a, )) 9. Conclude with the following: Theorem 118 Let n ≥ 1 and Ω, Ω be open in Rn . Let φ : Ω → Ω be a C 1 -diffeomorphism and ψ = φ−1 . Then, for all a ∈ Ω , we have: lim ↓↓0 φ(dx|Ω )(B (a, )) = |J (ψ )(a)| dx|Ω (B (a, )) where J (ψ )(a) is the jacobian of ψ at a, B (a, ) is the open ball in Rn , and dx|Ω , dx|Ω are the lebesgue measures on Ω and Ω respectively. Tutorial 18: The Jacobian Formula 32 Exercise 23. Let n ≥ 1 and Ω, Ω be open in Rn . Let φ : Ω → Ω be a C 1 -diffeomorphism and ψ = φ−1 . 1. Let K ⊆ Ω be a compact subset of Ω such that dx|Ω (K ) = 0. Given > 0, show the existence of V open in Ω , such that K ⊆ V ⊆ Ω , and dx|Ω (V ) ≤ . 2. Explain why V is also open in Rn . 3. Show that M = supx∈K dψ (x) < +∞. 4. For all x ∈ K , show there is x > 0 such that B (x, x ) ⊆ V , and for all h ∈ Rn with h ≤ 3 x , we have x + h ∈ Ω , and: ψ (x + h) − ψ (x) ≤ (M + 1) h 5. Show that for all x ∈ K , B (x, 3 x ) ⊆ Ω , and: ψ (B (x, 3 x )) ⊆ B (ψ (x), 3(M + 1) x ) 6. Show that ψ (B (x, 3 x )) = {φ ∈ B (x, 3 x )}, for all x ∈ K . Tutorial 18: The Jacobian Formula 33 7. Show the existence of {x1 , . . . , xp } ⊆ K , (p ≥ 0), such that: K ⊆ B (x1 , x1 ) ∪ . . . ∪ B (xp , xp ) 8. Show the existence of S ⊆ {1, . . . , p} such that the B (xi , are pairwise disjoint for i ∈ S , and: K⊆ B (xi , 3 xi )’s xi ) i∈S 9. Show that {φ ∈ K } ⊆ ∪i∈S B (ψ (xi ), 3(M + 1) 10. Show that φ(dx|Ω )(K ) ≤ i∈S xi ). 3n (M + 1)n dx(B (xi , xi )). 11. Show that φ(dx|Ω )(K ) ≤ 3n (M + 1)n dx(V ). 12. Show that φ(dx|Ω )(K ) ≤ 3n (M + 1)n . 13. Conclude that φ(dx|Ω )(K ) = 0. 14. Show that φ(dx|Ω ) is a locally finite measure on (Ω , B (Ω )). Tutorial 18: The Jacobian Formula 34 15. Let B ∈ B (Ω ) be such that dx|Ω (B ) = 0. Show that: φ(dx|Ω )(B ) = sup{φ(dx|Ω )(K ) : K ⊆ B , K compact } 16. Show that φ(dx|Ω )(B ) = 0. 17. Conclude with the following: Theorem 119 Let n ≥ 1, Ω, Ω be open in Rn , and φ : Ω → Ω be a C 1 -diffeomorphism. Then, the image measure φ(dx|Ω ), by φ of the lebesgue measure on Ω, is absolutely continuous with respect to dx|Ω , the lebesgue measure on Ω , i.e.: φ(dx|Ω ) << dx|Ω Tutorial 18: The Jacobian Formula 35 Exercise 24. Let n ≥ 1 and Ω, Ω be open in Rn . Let φ : Ω → Ω be a C 1 -diffeomorphism and ψ = φ−1 . 1. Explain why there exists a sequence (Vp )p≥1 of open sets in Ω , such that Vp ↑ Ω and for all p ≥ 1, the closure of Vp in Ω , i.e. ¯ VpΩ , is compact. ¯ ¯ 2. Show that each Vp is also open in Rn , and that VpΩ = Vp . 3. Show that φ(dx|Ω )(Vp ) < +∞, for all p ≥ 1. 4. Show that dx|Ω and φ(dx|Ω ) are two σ -finite measures on Ω . 5. Show there is h : (Ω , B (Ω )) → (R+ , B (R+ )) measurable, with: ∀B ∈ B (Ω ) , φ(dx|Ω )(B ) = B hdx|Ω 6. For all p ≥ 1, we define hp = h1Vp , and we put: ˜ ∀x ∈ Rn , hp (x) = hp (x) 0 if if x∈Ω x∈Ω Tutorial 18: The Jacobian Formula 36 Show that: ˜ hp dx = Rn Ω hp dx|Ω = φ(dx|Ω )(Vp ) < +∞ ˜ and conclude that hp ∈ L1 (Rn , B (Rn ), dx). R 7. Show the existence of some N ∈ B (Rn ), such that dx(N ) = 0 and for all x ∈ N c and p ≥ 1, we have: 1 ↓↓0 dx(B (x, )) ˜ hp dx ˜ hp (x) = lim B (x, ) 8. Put N = N ∩ Ω . Show that N ∈ B (Ω ) and dx|Ω (N ) = 0. 9. Let x ∈ Ω and p ≥ 1 be such that x ∈ Vp . Show that if > 0 is such that B (x, ) ⊆ Vp , then dx(B (x, )) = dx|Ω (B (x, )), and: ˜ hp dx = B (x, ) 1B (x, ) ˜ p dx = h Rn 1B (x, ) hp dx|Ω Ω Tutorial 18: The Jacobian Formula 37 10. Show that: Ω 1B (x, )hp dx|Ω = Ω 1B (x, )hdx|Ω = φ(dx|Ω )(B (x, )) 11. Show that for all x ∈ Ω \ N , we have: φ(dx|Ω )(B (x, )) ↓↓0 dx|Ω (B (x, )) h(x) = lim 12. Show that h = |J (ψ )| dx|Ω -a.s. and conclude with the following: Theorem 120 Let n ≥ 1 and Ω, Ω be open in Rn . Let φ : Ω → Ω be a C 1 -diffeomorphism and ψ = φ−1 . Then, the image measure by φ of the lebesgue measure on Ω, is equal to the measure on (Ω , B (Ω )) with density |J (ψ )| with respect to the lebesgue measure on Ω , i.e.: φ(dx|Ω ) = |J (ψ )|dx|Ω Tutorial 18: The Jacobian Formula 38 Exercise 25. Prove the following: Theorem 121 (Jacobian Formula 1) Let n ≥ 1 and φ : Ω → Ω be a C 1 -diffeomorphism where Ω, Ω are open in Rn . Let ψ = φ−1 . Then, for all f : (Ω , B (Ω )) → [0, +∞] non-negative and measurable: f ◦ φ dx|Ω = Ω f |J (ψ )|dx|Ω Ω and: (f ◦ φ)|J (φ)|dx|Ω = Ω Exercise 26. Prove the following: f dx|Ω Ω Tutorial 18: The Jacobian Formula 39 Theorem 122 (Jacobian Formula 2) Let n ≥ 1 and φ : Ω → Ω be a C 1 -diffeomorphism where Ω, Ω are open in Rn . Let ψ = φ−1 . Then, for all measurable map f : (Ω , B (Ω )) → (C, B (C)), we have the equivalence: f ◦ φ ∈ L1 (Ω, B (Ω), dx|Ω ) ⇔ f |J (ψ )| ∈ L1 (Ω , B (Ω ), dx|Ω ) C C in which case: Ω f ◦ φ dx|Ω = Ω f |J (ψ )|dx|Ω and, furthermore: (f ◦ φ)|J (φ)| ∈ L1 (Ω, B (Ω), dx|Ω ) ⇔ f ∈ L1 (Ω , B (Ω ), dx|Ω ) C C in which case: (f ◦ φ)|J (φ)|dx|Ω = Ω f dx|Ω Ω Tutorial 18: The Jacobian Formula 40 Exercise 27. Let f : R2→[0, +∞], with f (x, y ) = exp(−(x2 + y 2 )/2). 1. Show that: +∞ f (x, y )dxdy = 2 e −u 2 /2 du −∞ R2 2. Define: ∆1 = {(x, y ) ∈ R2 : x > 0 , y > 0} ∆2 = {(x, y ) ∈ R2 : x < 0 , y > 0} and let ∆3 and ∆4 be the other two open quarters of R2 . Show: f (x, y )dxdy = R2 f (x, y )dxdy ∆1 ∪...∪∆4 3. Let Q : R2 → R2 be defined by Q(x, y ) = (−x, y ). Show that: f ◦ Q−1 (x, y )dxdy f (x, y )dxdy = ∆1 ∆2 Tutorial 18: The Jacobian Formula 41 4. Show that: f (x, y )dxdy = 4 R2 f (x, y )dxdy ∆1 5. Let D1 =]0, +∞[×]0, π/2[⊆ R2 , and define φ : D1 → ∆1 by: ∀(r, θ) ∈ D1 , φ(r, θ) = (r cos θ, r sin θ) Show that φ is a bijection and that ψ = φ−1 is given by: ∀(x, y ) ∈ ∆1 , ψ (x, y ) = ( x2 + y 2 , arctan(y/x)) 6. Show that φ is a C 1 -diffeomorphism, with: ∀(r, θ) ∈ D1 , dφ(r, θ) = cos θ sin θ −r sin θ r cos θ and: ∀(x, y ) ∈ ∆1 , dψ (x, y ) = √ x x 2 +y 2 −y x 2 +y 2 √ y x 2 +y 2 x x 2 +y 2 Tutorial 18: The Jacobian Formula 42 7. Show that J (φ)(r, θ) = r, for all (r, θ) ∈ D1 . 8. Show that J (ψ )(x, y ) = 1/( x2 + y 2 ), for all (x, y ) ∈ ∆1 . 9. Show that: f (x, y )dxdy = ∆1 π 2 10. Prove the following: Theorem 123 We have: 1 √ 2π +∞ −∞ e −u 2 /2 du = 1 Tutorial 19: Fourier Transform 1 19. Fourier Transform Exercise 1. We define the maps ψ : R2 → C and φ : R → C: ∀(u, x) ∈ R2 , ψ (u, x) = eiux−x ∀u ∈ R , φ(u) = 2 /2 +∞ ψ (u, x)dx −∞ 1. Show that for all u ∈ R, the map x → ψ (u, x) is measurable. 2. Show that for all u ∈ R, we have: +∞ −∞ |ψ (u, x)|dx = √ 2π < +∞ and conclude that φ is well defined. 3. Let u ∈ R and (un )n≥1 be a sequence in R converging to u. Show that φ(un ) → φ(u) and conclude that φ is continuous. Tutorial 19: Fourier Transform 2 4. Show that: +∞ xe−x 2 /2 dx = 1 0 5. Show that for all u ∈ R, we have: +∞ −∞ ∂ψ (u, x) dx = 2 < +∞ ∂u 6. Let a, b ∈ R, a < b. Show that: b eib − eia = ieix dx a 7. Let a, b ∈ R, a < b. Show that: |eib − eia | ≤ |b − a| 8. Let a, b ∈ R, a = b. Show that for all x ∈ R: 2 ψ (b, x) − ψ (a, x) ≤ |x|e−x /2 b−a Tutorial 19: Fourier Transform 3 9. Let u ∈ R and (un )n≥1 be a sequence in R converging to u, with un = u for all n. Show that: +∞ φ(un ) − φ(u) = n→+∞ un − u lim −∞ ∂ψ (u, x)dx ∂u 10. Show that φ is differentiable with: ∀u ∈ R , φ (u) = +∞ −∞ ∂ψ (u, x)dx ∂u 11. Show that φ is of class C 1 . 12. Show that for all (u, x) ∈ R2 , we have: ∂ψ ∂ψ (u, x) = −uψ (u, x) − i (u, x) ∂u ∂x 13. Show that for all u ∈ R: +∞ −∞ ∂ψ (u, x) dx < +∞ ∂x Tutorial 19: Fourier Transform 4 14. Let a, b ∈ R, a < b. Show that for all u ∈ R: b ψ (u, b) − ψ (u, a) = a ∂ψ (u, x)dx ∂x 15. Show that for all u ∈ R: +∞ −∞ ∂ψ (u, x)dx = 0 ∂x 16. Show that for all u ∈ R: φ (u) = −uφ(u) Exercise 2. Let S be the set of functions defined by: S = {h : h ∈ C 1 (R, R) , ∀u ∈ R , h (u) = −uh(u)} 1. Let φ be as in ex. (1). Show that Re(φ) and Im(φ) lie in S . Tutorial 19: Fourier Transform 5 2. Given h ∈ S , we define g : R → R, by: 2 ∀u ∈ R , g (u) = h(u)eu /2 Show that g is of class C 1 with g = 0. 3. Let a, b ∈ R, a < b. Show the existence of c ∈]a, b[, such that: g (b) − g (a) = g (c)(b − a) 4. Conclude that for all h ∈ S , we have: ∀u ∈ R , h(u) = h(0)e−u 2 /2 5. Prove the following: Theorem 124 For all u ∈ R, we have: 1 √ 2π +∞ −∞ eiux−x 2 /2 dx = e−u 2 /2 Tutorial 19: Fourier Transform 6 Definition 135 Let µ1 , . . . , µp be complex measures on Rn ,1 where n, p ≥ 1. We call convolution of µ1 , . . . , µp , denoted µ1 . . . µp , the image measure of the product measure µ1 ⊗ . . . ⊗ µp by the measurable map S : (Rn )p → Rn defined by: S (x1 , . . . , xp ) = x1 + . . . + xp In other words, µ1 . . . µp is the complex measure on Rn , defined by: µ1 . . . µp = S (µ1 ⊗ . . . ⊗ µp ) Exercise 3. Let µ, ν be complex measures on Rn . 1. Show that for all B ∈ B (Rn ): µ ν (B ) = 1 An 1B (x R n ×R n + y )dµ ⊗ ν (x, y ) obvious shortcut to saying ’complex measures on (Rn , B(Rn ))’. Tutorial 19: Fourier Transform 7 2. Show that for all B ∈ B (Rn ): 1B (x + y )dµ(x) dν (y ) µ ν (B ) = Rn Rn 3. Show that for all B ∈ B (Rn ): 1B (x + y )dν (x) dµ(y ) µ ν (B ) = Rn 4. Show that µ ν = ν Rn µ. 5. Let f : Rn → C be bounded and measurable. Show that: f dµ ν = Rn f (x + y )dµ ⊗ ν (x, y ) R n ×R n Tutorial 19: Fourier Transform 8 Exercise 4. Let µ, ν be complex measures on Rn . Given B ⊆ Rn and x ∈ Rn , we define B − x = {y ∈ Rn , y + x ∈ B }. 1. Show that for all B ∈ B (Rn ) and x ∈ Rn , B − x ∈ B (Rn ). 2. Show x → µ(B − x) is measurable and bounded, for B ∈ B (Rn ). 3. Show that for all B ∈ B (Rn ): µ(B − x)dν (x) µ ν (B ) = Rn 4. Show that for all B ∈ B (Rn ): ν (B − x)dµ(x) µ ν (B ) = Rn Tutorial 19: Fourier Transform 9 Exercise 5. Let µ1 , µ2 , µ3 be complex measures on Rn . 1. Show that for all B ∈ B (Rn ): µ1 (µ2 µ3 )(B ) = 1B (x R n ×R n + y )dµ1 ⊗ (µ2 µ3 )(x, y ) 2. Show that for all B ∈ B (Rn ) and x ∈ Rn : 1B (x + y )dµ2 µ3 (y ) = Rn 1B (x R n ×R n + y + z )dµ2 ⊗ µ3 (y, z ) 3. Show that for all B ∈ B (Rn ): µ1 (µ2 µ3 )(B ) = 1B (x + y R n ×R n ×R n + z )dµ1 ⊗ µ2 ⊗ µ3 (x, y, z ) 4. Show that µ1 (µ2 µ3 ) = µ1 µ2 µ3 = (µ1 µ2 ) µ3 Tutorial 19: Fourier Transform 10 Definition 136 Let n ≥ 1 and a ∈ Rn . We define δa : B (Rn ) → R+ : ∀B ∈ B (Rn ) , δa (B ) = 1B (a) δa is called the dirac probability measure on Rn , centered in a. Exercise 6. Let n ≥ 1 and a ∈ Rn . 1. Show that δa is indeed a probability measure on Rn . 2. Show for all f : Rn → [0, +∞] non-negative and measurable: f dδa = f (a) Rn 3. Show if f : Rn → C is measurable, f ∈ L1 (Rn , B (Rn ), δa ) and: C f dδa = f (a) Rn Tutorial 19: Fourier Transform 11 4. Show that for any complex measure µ on Rn : µ δ0 = δ0 µ = µ 5. Let τa (x) = a + x define the translation of vector a in Rn . Show that for any complex measure µ on Rn : µ δa = δa µ = τa (µ) Exercise 7. Let n ≥ 1 and µ, ν be complex measures on Rn . We assume that ν << dx, i.e. that ν is absolutely continuous with respect to the lebesgue measure on Rn . 1. Show there is f ∈ L1 (Rn , B (Rn ), dx), such that ν = C 2. Show that for all B ∈ B (Rn ), we have: ν (B − x)dµ(x) µ ν (B ) = Rn f dx. Tutorial 19: Fourier Transform 12 3. Show that for all B ∈ B (Rn ) and x ∈ Rn : 1B (y )f (y − x)dy ν (B − x) = Rn 4. Show that for all B ∈ B (Rn ) the map: (x, y ) → 1B (y )f (y − x) lies in L1 (R × R , B (Rn ) ⊗ B (Rn ), µ ⊗ dy ). C n n 5. Show that for all B ∈ B (Rn ), we have: f (y − x)dµ(x) dy µ ν (B ) = B Rn 6. Given y ∈ Rn , we define: f (y − x)dµ(x) g (y ) = Rn Show that g (y ) is well-defined for dy -almost all y ∈ Rn . Tutorial 19: Fourier Transform 13 7. Define an element g of L1 (Rn , B (Rn ), dx), with g = g dx − a.s. ¯ ¯ C 8. Show that µ ν is absolutely continuous w.r. to the lebesgue measure on Rn , with density g . Theorem 125 Let µ, ν be two complex measures on Rn , n ≥ 1. If ν << dx, i.e. ν is absolutely continuous with respect to the lebesgue , measure on Rn , with density f ∈ L1 (Rn B (Rn), dx), then the convoC lution µ ν = ν µ is itself absolutely continuous with respect to the lebesgue measure on Rn , with density: f (y − x)dµ(x) , dy − a.s. g (y ) = Rn In other words, µ ν = ν µ= g dx. Exercise 8. Further to theorem (125), show that if µ = some h ∈ L1 (Rn , B (Rn ), dx), then: C f (y − x)h(x)dx , dy − a.s. g (y ) = Rn hdx for Tutorial 19: Fourier Transform 14 Definition 137 Let µ be a complex measure on (Rn , B (Rn )), n ≥ 1. We call fourier transform of µ, the map F µ : Rn → C defined by: ∀u ∈ Rn , F µ(u) = ei u,x dµ(x) Rn where ·, · is the usual inner-product in Rn . Exercise 9. Further to definition (137): 1. Show that F µ is well-defined. b 2. Show that F µ ∈ CC (Rn ), i.e F µ is continuous and bounded. 3. Show that for all a, u ∈ Rn , we have ∀u ∈ Rn , F δa (u) = ei 4. Let µ be the probability measure on (R, B (R)) defined by: 1 ∀B ∈ B (R) , µ(B ) = √ 2π Show that F µ(u) = e−u 2 /2 e −x B , for all u ∈ R. 2 /2 dx u,a . Tutorial 19: Fourier Transform 15 Exercise 10. Let µ1 , . . . , µp be complex measures on Rn , n, p ≥ 1. 1. Show that for all u ∈ Rn , we have: F (µ1 . . . µp )(u) = ei u,x1 +...+xp dµ1 ⊗ . . . ⊗ µp (x1 , . . . , xp ) (R n )p 2. Show that F (µ1 . . . µp ) = Πp=1 F µj . j Exercise 11. Let n ≥ 1, σ > 0 and gσ : Rn → R+ defined by: 2 2 1 e− x /2σ ∀x ∈ Rn , gσ (x) = n n (2π ) 2 σ 1. Show that: gσ (x)dx = 1 Rn 2. Show that for all u ∈ Rn , we have: gσ (x)ei Rn u,x dx = e−σ 2 u 2 /2 Tutorial 19: Fourier Transform 16 3. Show that Pσ = gσ dx is a probability on Rn with fourier transform: 2 2 ∀u ∈ Rn , F Pσ (u) = e−σ u /2 4. Show that for all x ∈ Rn , we have: gσ (x) = 1 (2π )n ei x,u −σ2 u 2 /2 du Rn Exercise 12. Further to ex. (11), let µ be a complex measure on Rn . 1. Show that µ Pσ = φσ dx where: gσ (x − y )dµ(y ) , dx − a.s. φσ (x) = Rn 2. Show that we also have: gσ (y − x)dµ(y ) , dx − a.s. φσ (x) = Rn Tutorial 19: Fourier Transform 17 3. Show that: φσ (x) = 1 (2π )n ei Rn y −x,u −σ2 u 2 /2 du dµ(y ) , dx − a.s. Rn 4. Show that: φσ (x) = 1 (2π )n e −i x,u −σ2 u 2 /2 (F µ)(u)du Rn 5. Show that if µ, ν are two complex measures on Rn such that F µ = F ν , then for all σ > 0, we have µ Pσ = ν Pσ . Definition 138 Let (Ω, T ) be a topological space. Let (µk )k≥1 be a sequence of complex measures on (Ω, B (Ω)). We say that the sequence (µk )k≥1 narrowly converges to a complex measure µ on (Ω, B (Ω)), and we write µk → µ narrowly, if and only if: b ∀f ∈ CR (Ω) , lim k→+∞ f dµk = f dµ Tutorial 19: Fourier Transform 18 Exercise 13. Further to definition (138): 1. Show that µk → µ narrowly, is equivalent to: b ∀f ∈ CC (Ω) , lim k→+∞ f dµk = f dµ 2. Show that if (Ω, T ) is metrizable and ν is a complex measure on (Ω, B (Ω)) such that µk → µ and µk → ν narrowly, then µ = ν . Theorem 126 On a metrizable topological space, the narrow limit when it exists, of any sequence of complex measures, is unique. Exercise 14. 1. Show that on (R, B (R)), we have δ1/n → δ0 narrowly. 2. Show there is B ∈ B (R), such that δ1/n (B ) → δ0 (B ). Tutorial 19: Fourier Transform 19 Exercise 15. Let n ≥ 1. Given σ > 0, let Pσ be the probability measure on (Rn , B (Rn )) defined as in ex. (11). Let (σk )k≥1 be a sequence in R+ such that σk > 0 and σk → 0. b 1. Show that for all f ∈ CR (Rn ), we have: f (x)gσk (x)dx = Rn 1 n (2π ) 2 f (σk x)e− Rn b 2. Show that for all f ∈ CR (Rn ), we have: f (x)gσk (x)dx = f (0) lim k→+∞ Rn 3. Show that Pσk → δ0 narrowly. x 2 /2 dx Tutorial 19: Fourier Transform 20 Exercise 16. Let µ, ν be two complex measures on Rn . Let (νk )k≥1 be a sequence of complex measures on Rn , which narrowly converges b to ν . Let f ∈ CR (Rn ), and φ : Rn → R be defined by: ∀y ∈ Rn , φ(y ) = f (x + y )dµ(x) Rn 1. Show that: f dµ νk = Rn f (x + y )dµ ⊗ νk (x, y ) R n ×R n 2. Show that: f dµ νk = Rn φdνk Rn b 3. Show that φ ∈ CC (Rn ). 4. Show that: φdνk = lim k→+∞ Rn φdν Rn Tutorial 19: Fourier Transform 21 5. Show that: f dµ νk = lim k→+∞ Rn f dµ ν Rn 6. Show that µ νk → µ ν narrowly. Theorem 127 Let µ, ν be two complex measures on Rn , n ≥ 1. Let (νk )k≥1 be a sequence of complex measures on Rn . Then: νk → ν narrowly ⇒ µ νk → µ ν narrowly Exercise 17. Let µ, ν be two complex measures on Rn , such that F µ = F ν . For all σ > 0, let Pσ be the probability measure on (Rn , B (Rn )) as defined in ex. (11). Let (σk )k≥1 be a sequence in R+ such that σk > 0 and σk → 0. 1. Show that µ Pσk = ν Pσk , for all k ≥ 1. 2. Show that µ Pσk → µ δ0 narrowly. Tutorial 19: Fourier Transform 22 3. Show that (µ Pσk )k≥1 narrowly converges to both µ and ν . 4. Prove the following: Theorem 128 Let µ, ν be two complex measures on Rn . Then: Fµ = Fν ⇒ µ=ν i.e. the fourier transform is an injective mapping on M 1 (Rn , B (Rn )). Definition 139 Let (Ω, F , P ) be a probability space. Given n ≥ 1, and a measurable map X : (Ω, F ) → (Rn , B (Rn )), the mapping φX defined as: ∀u ∈ Rn , φX (u) = E [ei u,X ] is called the characteristic function2 of the random variable X . 2 Do not confuse with the characteristic function 1A of a set A, definition (39). Tutorial 19: Fourier Transform 23 Exercise 18. Further to definition (139): 1. Show that φX is well-defined, bounded and continuous. 2. Show that we have: ∀u ∈ Rn , φX (u) = ei u,x dX (P )(x) Rn 3. Show φX is the fourier transform of the image measure X (P ). 4. Show the following: Theorem 129 Let X, Y : (Ω, F ) → (Rn , B (Rn )), n ≥ 1, be two random variables on a probability space (Ω, F , P ). If X and Y have the same characteristic functions, i.e. ∀u ∈ Rn , E [ei u,X = E [ei u,Y then X and Y have the same distributions, i.e. ∀B ∈ B (Rn ) , P ({X ∈ B }) = P ({Y ∈ B }) Tutorial 19: Fourier Transform 24 Definition 140 Let n ≥ 1. Given α = (α1 , . . . , αn ) ∈ Nn , we define the modulus of α, denoted |α|, by |α| = α1 + . . . + αn . Given x ∈ Rn and α ∈ Nn , we put: xα = xα1 . . . xαn n 1 α where it is understood that xj j = 1 whenever αj = 0. Given a map f : U → C, where U is an open subset of Rn , we denote ∂ α f the |α|-th partial derivative, when it exists: ∂ αf = ∂ |α| f . . . ∂xαn n ∂xα1 1 Note that ∂ α f = f , whenever |α| = 0. Given k ≥ 0, we say that f is of class C k , if and only if for all α ∈ Nn with |α| ≤ k , ∂ α f exists and is continuous on U . Exercise 19. Explain why def. (140) is consistent with def. (130). Tutorial 19: Fourier Transform 25 Exercise 20. Let µ be a complex measure on Rn , and α ∈ Nn , with: |xα |d|µ|(x) < +∞ (1) Rn Let xα µ the complex measure on Rn defined by xα µ = xα dµ. 1. Explain why the above integral (1) is well-defined. 2. Show that xα µ is a well-defined complex measure on Rn . 3. Show that the total variation of xα µ is given by: ∀B ∈ B (Rn ) , |xα µ|(B ) = |xα |d|µ|(x) B 4. Show that the fourier transform of xα µ is given by: ∀u ∈ Rn , F (xα µ)(u) = xα ei Rn u,x dµ(x) Tutorial 19: Fourier Transform 26 Exercise 21. Let µ be a complex measure on Rn . Let β ∈ Nn with |β | = 1, and: |xβ |d|µ|(x) < +∞ Rn Let xβ µ be the complex measure on Rn defined as in ex. (20). 1. Show that there is j ∈ Nn with xβ = xj for all x ∈ Rn . 2. Show that for all u ∈ Rn , ∂F µ ∂uj (u) ∂F µ (u) = i ∂uj exists and that we have: xj ei u,x dµ(x) Rn 3. Conclude that ∂ β F µ exists and that we have: ∂ β F µ = iF (xβ µ) 4. Explain why ∂ β F µ is continuous. Tutorial 19: Fourier Transform 27 Exercise 22. Let µ be a complex measure on Rn . Let k ≥ 0 be an integer. We assume that for all α ∈ Nn , we have: |xα |d|µ|(x) < +∞ |α| ≤ k ⇒ (2) Rn In particular, if |α| ≤ k , the measure xα µ of ex. (20) is well-defined. We claim that for all α ∈ Nn with |α| ≤ k , ∂ α F µ exists, and: ∂ α F µ = i|α| F (xα µ) 1. Show that if k = 0, then the property is obviously true. We assume the property is true for some k ≥ 0, and that the above integrability condition (2) holds for k + 1. 2. Let α ∈ Nn be such that |α | ≤ k + 1. Explain why if |α | ≤ k , then ∂ α F µ exists, with: ∂ α F µ = i|α | F (xα µ) 3. We assume that |α | = k + 1. Show the existence of α, β ∈ Nn such that α + β = α , |α| = k and |β | = 1. Tutorial 19: Fourier Transform 28 4. Explain why ∂ α F µ exists, and: ∂ α F µ = i|α| F (xα µ) 5. Show that: |xβ |d|xα µ|(x) < +∞ Rn 6. Show that ∂ β F (xα µ) exists, with: ∂ β F (xα µ) = iF (xβ (xα µ)) 7. Show that ∂ β (∂ α F µ) exists, with: ∂ β (∂ α F µ) = i|α|+1 F (xβ (xα µ)) 8. Show that xβ (xα µ) = xα µ. 9. Conclude that the property is true for k + 1. 10. Show the following: Tutorial 19: Fourier Transform 29 Theorem 130 Let µ be a complex measure on Rn , n ≥ 1. Let k ≥ 0 be an integer such that for all α ∈ Nn with |α| ≤ k , we have: |xα |d|µ|(x) < +∞ Rn Then, the fourier transform F µ is of class C k on Rn , and for all α ∈ Nn with |α| ≤ k , we have: ∀u ∈ Rn , ∂ α F µ(u) = i|α| xα ei u,x Rn In particular: xα dµ(x) = i−|α| ∂ α F µ(0) Rn dµ(x) Tutorial 20: Gaussian Measures 1 20. Gaussian Measures Mn (R) is the set of all n × n-matrices with real entries, n ≥ 1. Definition 141 A matrix M ∈ Mn (R) is said to be symmetric, if and only if M = M t . M is orthogonal, if and only if M is non-singular and M −1 = M t . If M is symmetric, we say that M is non-negative, if and only if: ∀u ∈ Rn , u, M u ≥ 0 Theorem 131 Let Σ ∈ Mn (R), n ≥ 1, be a symmetric and nonnegative real matrix. There exist λ1 , . . . , λn ∈ R+ and P ∈ Mn (R) orthogonal matrix, such that: 0 λ1 t .. Σ = P. .P . 0 λn In particular, there exists A ∈ Mn (R) such that Σ = A.At . Tutorial 20: Gaussian Measures 2 As a rare exception, theorem (131) is given without proof. Exercise 1. Given n ≥ 1 and M ∈ Mn (R), show that we have: ∀u, v ∈ Rn , u, M v = M t u, v Exercise 2. Let n ≥ 1 and m ∈ Rn . Let Σ ∈ Mn (R) be a symmetric and non-negative matrix. Let µ1 be the probability measure on R: 1 ∀B ∈ B (R) , µ1 (B ) = √ 2π e −x 2 /2 dx B Let µ = µ1 ⊗ . . . ⊗ µ1 be the product measure on Rn . Let A ∈ Mn (R) be such that Σ = A.At . We define the map φ : Rn → Rn by: ∀x ∈ Rn , φ(x) = Ax + m 1. Show that µ is a probability measure on (Rn , B (Rn )). 2. Explain why the image measure P = φ(µ) is well-defined. 3. Show that P is a probability measure on (Rn , B (Rn )). Tutorial 20: Gaussian Measures 3 4. Show that for all u ∈ Rn : F P (u) = ei u,φ(x) dµ(x) Rn 5. Let v = At u. Show that for all u ∈ Rn : F P (u) = ei u,m − v 2 /2 6. Show the following: Theorem 132 Let n ≥ 1 and m ∈ Rn . Let Σ ∈ Mn (R) be a symmetric and non-negative real matrix. There exists a unique complex measure on Rn , denoted Nn (m, Σ), with fourier transform: F Nn (m, Σ)(u) = ei u,x dNn (m, Σ)(x) = ei u,m − 1 u,Σu 2 Rn for all u ∈ Rn . Furthermore, Nn (m, Σ) is a probability measure. Tutorial 20: Gaussian Measures 4 Definition 142 Let n ≥ 1 and m ∈ Rn . Let Σ ∈ Mn (R) be a symmetric and non-negative real matrix. The probability measure Nn (m, Σ) on Rn defined in theorem (132) is called the n-dimensional gaussian measure or normal distribution, with mean m ∈ Rn and covariance matrix Σ. Exercise 3. Let n ≥ 1 and m ∈ Rn . Show that Nn (m, 0) = δm . Exercise 4. Let m ∈ Rn . Let Σ ∈ Mn (R) be a symmetric and non-negative real matrix. Let A ∈ Mn (R) be such that Σ = A.At . A map p : Rn → C is said to be a polynomial, if and only if, it is a finite linear complex combination of maps x → xα ,1 for α ∈ Nn . 1. Show that for all B ∈ B (R), we have: 1 N1 (0, 1)(B ) = √ 2π 1 See definition (140). e −x B 2 /2 dx Tutorial 20: Gaussian Measures 2. Show that: +∞ −∞ 5 |x|dN1 (0, 1)(x) < +∞ 3. Show that for all integer k ≥ 1: 1 √ 2π +∞ xk+1 e−x 2 /2 0 k dx = √ 2π +∞ xk−1 e−x 2 0 4. Show that for all integer k ≥ 0: +∞ −∞ |x|k dN1 (0, 1)(x) < +∞ 5. Show that for all α ∈ Nn : |xα |dN1 (0, 1) ⊗ . . . ⊗ N1 (0, 1)(x) < +∞ Rn /2 dx Tutorial 20: Gaussian Measures 6 6. Let p : Rn → C be a polynomial. Show that: |p(x)|dN1 (0, 1) ⊗ . . . ⊗ N1 (0, 1)(x) < +∞ Rn 7. Let φ : Rn → Rn be defined by φ(x) = Ax + m. Explain why the image measure φ(N1 (0, 1) ⊗ . . . ⊗ N1 (0, 1)) is well-defined. 8. Show that φ(N1 (0, 1) ⊗ . . . ⊗ N1 (0, 1)) = Nn (m, Σ). 9. Show if β ∈ Nn and |β | = 1, then x → φ(x)β is a polynomial. 10. Show that if α ∈ Nn and |α | = k +1, then φ(x)α = φ(x)α φ(x)β for some α, β ∈ Nn such that |α| = k and |β | = 1. 11. Show that the product of two polynomials is a polynomial. 12. Show that for all α ∈ Nn , x → φ(x)α is a polynomial. 13. Show that for all α ∈ Nn : |φ(x)α |dN1 (0, 1) ⊗ . . . ⊗ N1 (0, 1)(x) < +∞ Rn Tutorial 20: Gaussian Measures 7 14. Show the following: Theorem 133 Let n ≥ 1 and m ∈ Rn . Let Σ ∈ Mn (R) be a symmetric and non-negative real matrix. Then, for all α ∈ Nn , the map x → xα is integrable with respect to the gaussian measure Nn (m, Σ): |xα |dNn (m, Σ)(x) < +∞ Rn Exercise 5. Let m ∈ Rn . Let Σ = (σij ) ∈ Mn (R) be a symmetric and non-negative real matrix. Let j, k ∈ Nn . Let φ be the fourier transform of the gaussian measure Nn (m, Σ), i.e.: ∀u ∈ Rn , φ(u) = ei u,m − 1 u,Σu 2 1. Show that: xj dNn (m, Σ)(x) = i−1 Rn ∂φ (0) ∂uj Tutorial 20: Gaussian Measures 8 2. Show that: xj dNn (m, Σ)(x) = mj Rn 3. Show that: xj xk dNn (m, Σ)(x) = i−2 Rn ∂2φ (0) ∂uj ∂uk 4. Show that: xj xk dNn (m, Σ)(x) = σjk − mj mk Rn 5. Show that: (xj − mj )(xk − mk )dNn (m, Σ)(x) = σjk Rn Tutorial 20: Gaussian Measures 9 Theorem 134 Let n ≥ 1 and m ∈ Rn . Let Σ = (σij ) ∈ Mn (R) be a symmetric and non-negative real matrix. Let Nn (m, Σ) be the gaussian measure with mean m and covariance matrix Σ. Then, for all j, k ∈ Nn , we have: xj dNn (m, Σ)(x) = mj Rn and: (xj − mj )(xk − mk )dNn (m, Σ)(x) = σjk Rn Definition 143 Let n ≥ 1. Let (Ω, F , P ) be a probability space. Let X : (Ω, F ) → (Rn , B (Rn )) be a measurable map. We say that X is an n-dimensional gaussian or normal vector, if and only if its distribution is a gaussian measure, i.e. X (P ) = Nn (m, Σ) for some m ∈ Rn and Σ ∈ Mn (R) symmetric and non-negative real matrix. Exercise 6. Show the following: Tutorial 20: Gaussian Measures 10 Theorem 135 Let n ≥ 1. Let (Ω, F , P ) be a probability space. Let X : (Ω, F ) → Rn be a measurable map. Then X is a gaussian vector, if and only if there exist m ∈ Rn and Σ ∈ Mn (R) symmetric and non-negative real matrix, such that: ∀u ∈ Rn , E [ei u,X = ei u,m − 1 u,Σu 2 where ·, · is the usual inner-product on Rn . ¯ Definition 144 Let X : (Ω, F ) → R (or C) be a random variable on a probability space (Ω, F , P ). We say that X is integrable, if and only if we have E [|X |] < +∞. We say that X is square-integrable, if and only if we have E [|X |2 ] < +∞. Exercise 7. Further to definition (144), suppose X is C-valued. 1. Show X is integrable if and only if X ∈ L1 (Ω, F , P ). C 2. Show X is square-integrable, if and only if X ∈ L2 (Ω, F , P ). C Tutorial 20: Gaussian Measures 11 ¯ Exercise 8. Further to definition (144), suppose X is R-valued. 1. Show that X is integrable, if and only if X is P -almost surely equal to an element of L1 (Ω, F , P ). R 2. Show that X is square-integrable, if and only if X is P -almost surely equal to an element of L2 (Ω, F , P ). R Exercise 9. Let X, Y : (Ω, F ) → (R, B (R)) be two square-integrable random variables on a probability space (Ω, F , P ). 1. Show that both X and Y are integrable. 2. Show that XY is integrable 3. Show that (X −E [X ])(Y −E [Y ]) is a well-defined and integrable. Tutorial 20: Gaussian Measures 12 Definition 145 Let X, Y : (Ω, F ) → (R, B (R)) be two squareintegrable random variables on a probability space (Ω, F , P ). We define the covariance between X and Y , denoted cov (X, Y ), as: cov (X, Y ) = E [(X − E [X ])(Y − E [Y ])] We say that X and Y are uncorrelated if and only if cov (X, Y ) = 0. If X = Y , cov (X, Y ) is called the variance of X , denoted var(X ). Exercise 10. Let X, Y be two square integrable, real random variable on a probability space (Ω, F , P ). 1. Show that cov (X, Y ) = E [XY ] − E [X ]E [Y ]. 2. Show that var(X ) = E [X 2 ] − E [X ]2 . 3. Show that var(X + Y ) = var(X ) + 2cov (X, Y ) + var(Y ) 4. Show that X and Y are uncorrelated, if and only if: var(X + Y ) = var(X ) + var(Y ) Tutorial 20: Gaussian Measures 13 Exercise 11. Let X be an n-dimensional normal vector on some probability space (Ω, F , P ), with law Nn (m, Σ), where m ∈ Rn and Σ = (σij ) ∈ Mn (R) is a symmetric and non-negative real matrix. 1. Show that each coordinate Xj : (Ω, F ) → R is measurable. 2. Show that E [|X α |] < +∞ for all α ∈ Nn . 3. Show that for all j = 1, . . . , n, we have E [Xj ] = mj . 4. Show that for all j, k = 1, . . . , n, we have cov (Xj , Xk ) = σjk . Theorem 136 Let X be an n-dimensional normal vector on a probability space (Ω, F , P ), with law Nn (m, Σ). Then, for all α ∈ Nn , X α is integrable. Moreover, for all j, k ∈ Nn , we have: E [Xj ] = mj and: cov (Xj , Xk ) = σjk where (σij ) = Σ. Tutorial 20: Gaussian Measures 14 Exercise 12. Show the following: Theorem 137 Let X : (Ω, F ) → (R, B (R)) be a real random variable on a probability space (Ω, F , P ). Then, X is a normal random variable, if and only if it is square integrable, and: ∀u ∈ R , E [eiuX ] = eiuE [X ]− 2 u 1 2 var (X ) Exercise 13. Let X be an n-dimensional normal vector on a probability space (Ω, F , P ), with law Nn (m, Σ). Let A ∈ Md,n (R) be an d × n real matrix, (n, d ≥ 1). Let b ∈ Rn and Y = AX + b. 1. Show that Y : (Ω, F ) → (Rd , B (Rd )) is measurable. 2. Show that the law of Y is Nd (Am + b, A.Σ.At ) 3. Conclude that Y is an Rd -valued normal random vector. Tutorial 20: Gaussian Measures 15 Theorem 138 Let X be an n-dimensional normal vector with law Nn (m, Σ) on a probability space (Ω, F , P ), (n ≥ 1). Let d ≥ 1 and A ∈ Md,n (R) be an d × n real matrix. Let b ∈ Rd . Then, Y = AX + b is an d-dimensional normal vector, with law: Y (P ) = Nd (Am + b, A.Σ.At ) Exercise 14. Let X : (Ω, F ) → (Rn , B (Rn )) be a measurable map, where (Ω, F , P ) is a probability space. Show that if X is a gaussian vector, then for all u ∈ Rn , u, X is a normal random variable. Exercise 15. Let X : (Ω, F ) → (Rn , B (Rn )) be a measurable map, where (Ω, F , P ) is a probability space. We assume that for all u ∈ Rn , u, X is a normal random variable. 1. Show that for all j = 1, . . . , n, Xj is integrable. 2. Show that for all j = 1, . . . , n, Xj is square integrable. 3. Explain why given j, k = 1, . . . , n, cov (Xj , Xk ) is well-defined. Tutorial 20: Gaussian Measures 16 4. Let m ∈ Rn be defined by mj = E [Xj ], and u ∈ Rn . Show: E [ u, X ] = u, m 5. Let Σ = (cov (Xi , Xj )). Show that for all u ∈ Rn , we have: var( u, X ) = u, Σu 6. Show that Σ is a symmetric and non-negative n × n real matrix. 7. Show that for all u ∈ Rn : E [ei u,X = eiE [ u,X ]− 1 var ( u,X ) 2 8. Show that for all u ∈ Rn : E [ei u,X = ei 9. Show that X is a normal vector. 10. Show the following: u,m − 1 u,Σu 2 Tutorial 20: Gaussian Measures 17 Theorem 139 Let X : (Ω, F ) → (Rn , B (Rn )) be a measurable map on a probability space (Ω, F , P ). Then, X is an n-dimensional normal vector, if and only if, any linear combination of its coordinates is itself normal, or in other words u, X is normal for all u ∈ Rn . Exercise 16. Let (Ω, F ) = (R2 , B (R2 )) and µ be the probability on (R, B (R)) defined by µ = 1 (δ0 + δ1 ). Let P = N1 (0, 1) ⊗ µ, and 2 X, Y : (Ω, F ) → (R, B (R)) be the canonical projections defined by X (x, y ) = x and Y (x, y ) = y . 1. Show that P is a probability measure on (Ω, F ). 2. Explain why X and Y are measurable. 3. Show that X has the distribution N1 (0, 1). 4. Show that P ({Y = 0}) = P ({Y = 1}) = 1 . 2 5. Show that P (X,Y ) = P . Tutorial 20: Gaussian Measures 18 6. Show for all φ : (R2 , B (R2 )) → C measurable and bounded: E [φ(X, Y )] = 1 (E [φ(X, 0)] + E [φ(X, 1)]) 2 7. Let X1 = X and X2 be defined as: X2 = X 1{Y =0} − X 1{Y =1} Show that E [eiuX2 ] = e−u 2 /2 for all u ∈ R. 8. Show that X1 (P ) = X2 (P ) = N1 (0, 1). 9. Explain why cov (X1 , X2 ) is well-defined. 10. Show that X1 and X2 are uncorrelated. 11. Let Z = 1 (X1 + X2 ). Show that: 2 ∀u ∈ R , E [eiuZ ] = 2 1 (1 + e−u /2 ) 2 Tutorial 20: Gaussian Measures 19 12. Show that Z cannot be gaussian. 13. Conclude that although X1 , X2 are normally distributed, (and even uncorrelated), (X1 , X2 ) is not a gaussian vector. Exercise 17. Let n ≥ 1 and m ∈ Rn . Let Σ ∈ Mn (R) be a symmetric and non-negative real matrix. Let A ∈ Mn (R) be such that Σ = A.At . We assume that Σ is non-singular. We define pm,Σ : Rn → R+ by: ∀x ∈ Rn , pm,Σ (x) = 1 (2π ) n 2 1 det(Σ) 1. Explain why det(Σ) > 0. 2. Explain why det(Σ) = | det(A)|. 3. Explain why A is non-singular. e− 2 x−m,Σ−1 (x−m) Tutorial 20: Gaussian Measures 20 4. Let φ : Rn → Rn be defined by: ∀x ∈ Rn , φ(x) = A−1 (x − m) Show that for all x ∈ Rn , x − m, Σ−1 (x − m) = φ(x) 2 . 5. Show that φ is a C 1 -diffeomorphism. 6. Show that φ(dx) = | det(A)|dx. 7. Show that: pm,Σ (x)dx = 1 Rn 8. Let µ = pm,Σ dx. Show that: ∀u ∈ Rn , F µ(u) = 1 n (2π ) 2 ei u,Ax+m − x 2 /2 dx Rn 9. Show that the fourier transform of µ is therefore given by: ∀u ∈ Rn , F µ(u) = ei u,m − 1 u,Σu 2 Tutorial 20: Gaussian Measures 21 10. Show that µ = Nn (m, Σ). 11. Show that Nn (m, Σ) << dx, i.e. that Nn (m, Σ) is absolutely continuous w.r. to the lebesgue measure on Rn . Exercise 18. Let n ≥ 1 and m ∈ Rn . Let Σ ∈ Mn (R) be a symmetric and non-negative real matrix. We assume that Σ is singular. Let u ∈ Rn be such that Σu = 0 and u = 0. We define: B = {x ∈ Rn , u, x = u, m } Given a ∈ Rn , let τa : Rn → Rn be the translation of vector a. −1 1. Show B = τ−m (u⊥ ), where u⊥ is the orthogonal of u in Rn . 2. Show that B ∈ B (Rn ). 3. Explain why dx(u⊥ ) = 0. Is it important to have u = 0? 4. Show that dx(B ) = 0. Tutorial 20: Gaussian Measures 22 5. Show that φ : Rn → R defined by φ(x) = u, x , is measurable. 6. Explain why φ(Nn (m, Σ)) is a well-defined probability on R. 7. Show that for all α ∈ R, we have: F φ(Nn (m, Σ))(α) = eiα u,x dNn (m, Σ)(x) Rn 8. Show that φ(Nn (m, Σ)) is the dirac distribution on (R, B (R)) centered on u, m , i.e. φ(Nn (m, Σ)) = δ u,m . 9. Show that Nn (m, Σ)(B ) = 1. 10. Conclude that Nn (m, Σ) cannot be absolutely continuous with respect to the lebesgue measure on (Rn , B (Rn )). 11. Show the following: Tutorial 20: Gaussian Measures 23 Theorem 140 Let n ≥ 1 and m ∈ Rn . Let Σ ∈ Mn (R) be a symmetric and non-negative real matrix. Then, the gaussian measure Nn (m, Σ) is absolutely continuous with respect to the lebesgue measure on (Rn , B (Rn )), if and only if Σ is non-singular, in which case for all B ∈ B (Rn ), we have: Nn (m, Σ)(B ) = 1 (2π ) n 2 e− 2 1 det(Σ) B x−m,Σ−1 (x−m) dx ...
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