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real_analysis_I

real_analysis_I - Real Analysis course outline Denis...

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Real Analysis, course outline Denis Labutin 1 Measure theory I 1. Sigma algebras. Let A be a collection of subsets of some fixed set Ω . It is called a σ -algebra with the unit element Ω if (a) , Ω ∈ A ; (b) E ∈ A = c E ∈ A ; (c) E j ∈ A , j = 1 , 2 , . . . = [ j E j ∈ A . Prove that a σ -algebra A is closed under countable number of the set theoretic operations ( , , \ , ( · ) c ). 2. For a sequence of sets E j define lim sup j →∞ E j = { points which belong to infinitely many E j } , lim inf j →∞ E j = { points which belong to all but finitely many E j } . Thus lim sup E j lim inf E j . If lim sup E j = lim inf E j = E , then we write lim j →∞ E j = E. (a) Give an example of a sequence { E j } for which the inclusion is strict. (b) Show that lim sup j E j = \ j =1 [ k = j E k , lim inf j E j = [ j =1 \ k = j E k . (c) Show that if all E j are elements of a σ -algebra A , then lim sup j E j , lim inf j E j ∈ A . (d) Suppose that E 1 E 2 ⊃ · · · . Prove that lim sup j E j = lim inf j E j = \ j E j . 1
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(e) Find (and prove) the formula for lim sup E j , lim inf E j in the case E 1 E 2 ⊂ · · · . 3. Prove that the following collections of sets are σ -algebras. (a) Trivial σ -algebra {∅ , Ω } . (b) Bulean σ -algebra 2 Ω = { all subsets of Ω } . (c) For any E Ω the collection { E, c E, , Ω } (It is called the σ - algebra generated by the set E ). (d) Let D = { D 1 , D 2 , . . . } is a countable (disjoint) partition of Ω : Ω = [ j D j , D i D j = if i 6 = j. The family of all at most countable unions of elements of D (includ- ing ) is a σ -algebra. (e) The family of all E Ω such that either E is at most countable or c E is at most countable. 4. New σ -algebras from old. Prove the following statements. (a) Give an example showing that the union of two σ -algebras with the same unit element is not necessarily a σ -algebra. (b) If ( A 1 , Ω) and ( A 2 , Ω) are σ -algebras then A 1 ∩ A 2 is also a σ - algebra with the same unit element Ω . (c) Similarly, if ( A a , Ω) is a σ -algebra for any a A , then \ a A A a is also a σ -algebra with the same unit element Ω . (d) If ( A , Ω) is a σ -algebra, and E Ω , then A ∩ E = { A E : A ∈ A} is a σ -algebra with the unit E Ω = E . (e) If ( A , Ω) is a σ -algebra, and Ξ is any fixed set, then A × Ξ = { A × Ξ: A ∈ A} is a σ -algebra with the unit Ω × Ξ . One should think of Ξ as a group of dummy variables. 5. Property (b) is the basis for the following important construction. Theorem 1 Let C be any collection of subsets of Ω . Then there exists one and only one σ -algebra ( A , Ω) such that: (i) C ⊂ A . (ii) For any σ -algebra ( e A , Ω) containing C we have e A ⊃ A . It is called the σ -algebra generated by C and is denoted by σ ( C ) . 2
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For the proof just define σ ( C ) def = \ all σ - algebras , A α ): C⊂A α A α . Hence, the more careful notation should be σ Ω ( C ) .
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