Ch0 - Chapter 0 Measures, Integration, Convergence 1....

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Unformatted text preview: Chapter 0 Measures, Integration, Convergence 1. Measures Definitions Basic Examples Extension Theorem Completion 2. Measurable Functions and Integration Simple functions Monotone convergence theorem (MCT) Fatou’s lemma Dominated convergence theorem (DCT) Absolute continuity of the integral Induced measures Theorem of the unconscious statistician 3. Absolute Continuity, Radon-Nikodym Theorem, Fubini’s Theorem Absolute continuity Radon—Nikodym Theorem Fubini—Tonelli Theorem Chapter 0 Measures, Integration, Convergence 1 Measures Let Q be a fixed non—void set. Definition 1.1 ( fields, U—fields, monotone classes) A non—void class A of subsets of Q is called a: (i) field or algebra if A, B E A implies A U B E A and AC 6 A. (ii) U—field or U—algebra if A,A1,A2, . . . E A implies UfoAn E A and AC 6 A. (iii) monotone class if An is a monotone / (\) sequence in A implies UfoAn E A (FROAH E A). (iv) (Q,A) With A a U—field of subsets of Q is called a measurable space. Remark 1.1 A,B E A imply A H B E A for a field. (ii) A1, . . .,A,,, . . . E A implies flfileAn E A for a U—field. (iii) (1), Q E A for both a field and U—field. (iv) To prove that A is a field (U—field) it suffices to show that A is closed under complements and finite (countable) intersections. Proposition 1.1 Arbitrary intersections of fields (U—fields) ((monotone classes)) are fields (U—fields) ((monotone classes)). (ii) There exists a minimal field (0— field) ((monotone class)) 0(C) generated by any class of subsets of 9. (iii) a U—field is a monotone class and conversely if it is a field. Proof. (iii) (<2) ugozlAn : Ugo:1(UZ:1Ak) E Ufan where B,, /. Notation 1.1 If 9 is a set, 29 is the family of all subsets of Q. 20 is always a U—field. Example 1.1 If 9 : R, let BO consist of(1) together With all finite unions of disjoint intervals of the form U?:1(a,,1),], or U?:1(a,,1),] U (a,,+1,oo), (—oo, 1),,+1] U U?:1(a,,1),], With (1,, 1), E R. Then 80 is a field. 4 CHAPTER 0. MEASURES, INTEGRATION, CONVERGENCE Example 1.2 If 9 : (0,1], let BO consist on) together with all finite unions of disjoint intervals of the form U?:1(a,,b,], g a, < b, g 1. Then 80 is a field. But note that 80 does not contain intervals of the form [(1,1) or ((1,1)); however ((1,1)) : og°=1(a,b — Example 1.3 If 9 : R, let C : 80 of example 1.1, and let 8 be the U—field generated by 80; B : 0(80). 8 is a U—field which contains all intervals, open, closed or half—open. From real analysis, any open set 0 C R can be written as a countable union of (disjoint) open intervals: 0 : Ugo:1(an, bn). Thus 8 contains all open sets in R. This particular 8 E 81 is called the family of Borel sets. In fact, 8 : 0(0), where O is the collection of all open sets in R. Example 1.4 Suppose that Q is a metric space with metric ,0. Let (9 be the collection of open subsets of Q. The the U—field B : 0(0) is called the Borel U—field. In particular, for Q : Rk with the Euclidean metric p($,y) : |$ — y| : — yi|2}1/2, B E Bk E 0(0) is the U—field of Borel sets. Definition 1.2 A measure (finitely additive measure) is a function ,u : A —> [0, 00] such that ,u((Z)) : 0 and ME An) : ZMAn) for countable (finite) disjoint sequences An in A. (ii) A measure space is a triple (Q,A,,u) with A a U—field and ,u a measure. Definition 1.3 ,u is a finite measure if 11(9) < 00. (ii) ,u is a probability measure if ,u(Q) : 1. (iii) ,u is an infinite measure if ,u(Q) : 00. (iv) A measure ,u on a field (U—field) A is called U—finite if there exists a partition {Fn}n21 C A such that Q : F7, and ,u(Fn) < 00 for all n 2 1. (v) A probability space is a measure space (Q,A,,u) with ,u a probability measure. Definition 1.4 A measure ,u on (Q, A) is discrete if there are finitely or countably many points to, in Q and masses m, E [0, 00) such that ,u(A) : E m, for A E A. wiEA (ii) If ,u is defined on (9,29), 9 arbitrary, by ,u(A) : fiof points in A, ,u(A) : 00 if A is not finite, then ,u is called counting measure. Example 1.5 A discrete measure ,u on (Q,A) : (R1,81): x, : i, m, : 2i. (ii) A discrete measure ,u on (Q,A) : (Z+,QZ+): x, : 22,771, : 1/2'. (Z+ : {1,2, . . (iii) Counting measure on (R1,81); not a U—finite measure! (iv) Counting measure on (Z+, 22+). (v) A probability measure on Q, the rationals: With an enumeration of the rationals, let m, : 6/(7r2i2). Proposition 1.2 Let (Q,A,,u) be a measure space. (i) If {An}n21 C A with An C An+1 for all n, then ,u(U%°=1An) : limn_,OO ,u(An). (ii) If ,u(A1) < 00 and An 3 An+1 for all n, then ,u(fl,‘3,°:1An) : limn_,OO ,u(An). 1. MEASURES 5 Proof. M(U?°An) M(U?°(An \ 1471—1)) Where A0 :0 E 2 MAR \ An_1) by countable additivity 1 TL = ligl Zu<An\An_1) lim M(Bn) = M A1) — MflfoAn) by finite additivity, while on the other hand, li7m ,u(Bn) : liqrbn ,u(A1 \ An) : li7m{M(A1) — M(An)} by finite additivity = M(A1) — 11,? MAU- Combining these two equalities yield the conclusion of (ii). Definition 1.5 (i) li_mAn E U221 0:02,, Ag, E {w E Q : w 6 all but a finite number ofAzs} E [An mm]; (ii) EAR E 0,2021 Uzin Ag, E {w E 9 :w E infinitely manyAzs} E [An i.0.]. Remark 1.2 li_mAn C mAn; limAn E li_mAn provided li_mAn : EA” Proposition 1.3 Monotone / (\) An’s have lim An : UfoAn (2 {KOAR Example 1.6 Let A : B : 0(80) as in example 1.3. For B E 80, let ,u(B) E the sum of the lengths of intervals A E 80 composing B. Then ,u is a countably additive measure on 80. Can ,u be extended to B? The answers is yes, and depends on the following: Theorem 1.1 (Caratheodory Extension Theorem) A measure ,u on a field C can be extended to a measure on the minimal U—field U—field 0(C) over C. If ,u is U—finite on C, then the extension is unique and is also U—finite. Proof. See Billingsley (1986), pages 29 — 35 and 137 — 139. 6 CHAPTER 0. MEASURES, INTEGRATION, CONVERGENCE Example 1.7 (example 1.3, continued.) The extension of the countably additive measure ,u on 80 to 81 : 0(80), the Boreal U—field, is called Lebesgue measure; thus (R1,81,,u) where ,u is the extension of the Caratheodory extension theorem, is a measure space. The usual procedure is to complete 81 as follows. Definition 1.6 If (Q,A,,u) is a measure space such that B C A with A E A and ,u(A) : 0 implies B E A, then (Q,A,,u) is a complete measure space. If ,u(A) : 0, then A is called a null set. (Of course there can be non—empty null sets.) Exercise 1.1 Let (Q,A,,u) be a measure space. Define XE {AUN:AEA,N C Bfor someB EAsuch that ,u(B) :0} and let fi(A U N) E ,u(A). Then (9,],fi) is a complete measure space. Example 1.8 (example 1.3, continued.) Completing (R1,81,,u) where ,u :Lebesgue measure yields the complete measure space (R1,El,fi). 8—1 is called the U—field of Lebesgue sets. So far we know only a few measures. But we will now construct a whole batch of them; and they are just the ones most useful for probability theory. Definition 1.7 A measure ,u on R assigning finite values to finite intervals is called a Lebesgue — Stieltjes measure. Definition 1.8 A function F on R which is finite, increasing, and right continuous is called a generalized distribution function (generalized df). Fm, b] E m) — Fm) for —00 < a < b < 00 is called the increment function of the generalized df F. We identify generalized df’s having the same increment function. Theorem 1.2 (Correspondence theorem.) The relation ,u((a,b]):F(a,b] for —oo<a§b<oo establishes a one—to—one correspondence between Lebesgue—Stieltjes measures ,u on B : 81 and equivalence classes of generalized df’s. Proof. See Billingsley (1986), pages 147, 149 — 151. Definition 1.9 (Probability measures on R.) If ,u(Q) : 1, then ,u is called a probability dis— tribution or probability measure and is denoted by P. Definition 1.10 An /, right—continuous function F on R such that F(—oo) : 0 and F(oo) : 1 is a distribution function (df). Corollary 1 The relation P((a,b]):F(b)—F(a) for —oo<a§b<oo establishes a one—to—one correspondence between probability measures on R and df’s. 2. MEASURABLE FUNCTIONS AND INTEGRATION 7 2 Measurable Functions and Integration Let (SLA) be a measurable space. Let X denote a function7 X : 9 —> R. Definition 2.1 X : 9 —> R is measurable if [X E B] E X_1(B) : {w E Q : X(w) E B} E A for all B681. Definition 2.2 For A E A the indicator function of A is the function 1A(w) :{ iwaA ifuJEAC (ii) A simple function is X(w) E 1 3621141. (w) for Air : 9, AZ' 6 A, 962' E R. 2: (iii) An elementary function is X(w) E $21141. (w) for AZ' : 9, AZ' 6 A, 96¢ E R. Proposition 2.1 X is measurable if and only if X_1(C) E {X_1(C) : C E C} C A Where 0(C) : 8. Hence X is measurable if and only if X_1((907 00)) E [X > 90] E A for all 90 E R. Proof. (=>) This direction is trivial. (<2) X_1(B) : X_1(0'(C)) : 0(X_1(C)) since X‘1 preserves all set operations and since X_1(C) C A With A a U—field by hypothesis. Further7 U({(x, 00) : 90 E R}) : 81 since (a,b] : ((1.00) H (b,oo)c7 and 81 is generated by intervals of the form (a, b]. Note that the assertion of the propostion would work With (90, oo) replaced by any Of [$7OO)7 (—007xl7 (—007x)‘ Proposition 2.2 Suppose that {Xn} are measurable. Then so are supn Xn7 —Xn7 infn Xn7 EXR7 li_an7 and lim Xn. Proof. [811an > $l : UanTL > xii [—Xn > x] : [Xn < —x]; inf Xn : — supn(—Xn); EXR : infn(supk2n Xk); 11—an = —E(—Xn); limn Xn : EXR When lim Xn exists. Proposition 2.3 X is measurable if and only if it is the limit of a sequence of simple functions: Xn = _n1[X<—n] -|- 2 Proof. 2n 1[(k—1)/2"SX<k/2“] + ”1[X>n1- k:—n2"+1 (=>) The Xn’s exhibited above have |Xn(w) — X(w)| < 2‘” for |X(w)| < n. (<2) The exhibited Xn’s are simple7 converge to X7 and lim Xn is measurable by prop 2.2. Remark 2.1 If X Z 07 then 0 g Xn / X. 8 CHAPTER 0. MEASURES, INTEGRATION, CONVERGENCE Proposition 2.4 Let X,Y be measurable. Then X :: Y, XY, X/Y, X+ E X1[XZO], X‘ E —X1[XSO], |X|, g(X) for measurable g are all measurable. Proof. Let Xn, Yn be simple functions, X7, —> X, Yn —> Y. Then Xn :: Yn, XnYn, Xn/Yn are simple functions converging to X :: Y, XY, and X/Y, and hence the limits are measurable by prop 2.3. X‘l' and X‘ are easy by prop 2.3, and |X| : X‘l' —|— X‘. For g : R —> R measurable we have, for B E 81, (gX)_1(B) : _1(g_1(B)) : X_1( a Borel set ) since 9 is measurable X E A since X‘1 is measurable. Remark 2.2 Any continuous function g is measurable since g_1(8) : g_1(0((’))) : (fig—1(0)): 0( a subcollection of open sets ) C 8. Now let (Q,A,,u) be a measure space, and leet X,Y denote measurable functions from (Q,A) to (R,E), R E R U {:00}, E E 0(8 U {00} U {—oo}). CONVENTIONS:0-OO:0:OO-0,$-00:oo-$:ooif0<ac<00;00-00200. Definition 2.3 For X E 36,1,4, with x, Z 0, A, : Q, then fXdM : x,M(A,). (ii) For X Z 0, fXdM E limanndu where {Xn} is any 2 0, / sequence of simple functions, Xn —> X. (iii) For general X, fXdM E fX‘l'dM — fX‘dM if one of fX‘l'dM, fX‘dM is finite. (iv) If fXdM is finite, then X is integrable. JUSTIFICATION: See Loeve pages 120 — 123 or Billingsley (1986), page 176. Proposition 2.5 (Elementary properties.) Suppose that fXdM, deu, and fXdM —|— deM exist. Then: (i) I(X+Y)du = fXdH‘l'deMa chdM = CfXdu; (ii) X Z 0 implies fXdM Z 0; X Z Y implies fXdM Z deM; and X : Y a.e. implies fXdM : deM. (iii) (integrability). X is integrable if and only if |X| is integrable, and either implies that X is a.e. finite. |X| g Y with Y integrable implies X integrable; X and Y integrable implies that X —|— Y is integrable. Proof. (iii) That X is integrable if and only if fX‘l’dM and fX‘dM finite if and only if |X| integrable is easy. Now fX‘l'du < 00 implies X+ finite a.e.; if not, then ,u(A) > 0 where A E {w : X+(w) : 00}, and then fX‘l'dM Z fX+1Ad,u : oo -,u(A) : 00, a contradiction. Now 0 g X+ g Y, thus 0 g fX‘l’dM g deM < 00. Likewise fX‘du < oo. Theorem 2.1 (Monotone convergence theorem.) If 0 3 X7, / X, then andlu —> fXdM. Corollary 1 If X7, 2 0 then [220:1 Xndu : 220:1 andM. 2. MEASURABLE FUNCTIONS AND INTEGRATION 9 Proof. Note that 0 g Xk / Xk and apply the monotone convergence theorem. Theorem 2.2 (Fatou’s lemma.) If X7, 2 0 for all n, then fli_and,u g Xndu . Proof. Since X7, Z inka,, Xk E Y7, / li_an, it follows from the MCT that /li_and,u : /limYnd,u : lim/Yndu g li_m/Xnd,u. Definition 2.4 A sequence X7, converges almost everywhere (or converges a.e. for short), denoted X7, —>a.e. X, if X,,(w) —> X(w) for all to E 9 \ N where MN) : 0 (i.e. for a.e. to). Note that {Xn}, X, are all defined on one measure space (9,A). If ,u is a probability measure, ,u : P with P(9) : 1, we will write —>a.5. for —>a.e.. Proposition 2.6 Let {Xn}, X be finite measurable functions. Then [Xn —> X] : 020:1 U221 flgfanXm — XI < 1/k], and is a measurable set. Corollary 1 Let {Xn}, X be finite measurable functions. Then X7, —>a.e. X if and only if limit; U232. lle — XI 2 6) = 0 for all 6 > 0. If M9) < 00, X7, —>a.e. X if and only if M(U:,O:n[|Xm — X| Z 6) —> 0 as n —> 00 for all 6 > 0. Proof. First note that an —> ch : U20:1 fl720:1 Ufnoznlle — XI 2 1/kl E UzO=1Ak with Ak /; and Ag, : D” Em with Bnk \ in n. Applying prop 1.2 gives the result. 7121 Definition 2.5 (Convergence in measure; convergence in probability.) A sequence of fi— nite measurable functions X7, converge in measure to a measurable function X, denoted X,, —>,) X, if Mlan — XI 2 6]) —> 0 for all 6 > 0. If ,u is a probability measure, M9) : 1, call ,u : P, write X7, —>,, X, and say X7, converge in probability to X. Proposition 2.7 Let Xn’s be finite a.e. (i) If X,, —>,, X then there exist a subsequence such that Xnk —>a.e. X. (ii) If M9) < 00 and X7, —>a.e. X, then X,, —>,, X. Theorem 2.3 (Dominated Convergence Theorem) If |Xn| g Y a.e. with Y integrable, and if X,, —>,, X (or X,, —>a.e. X), then f |Xn — X|dlu —> 0 and limendM : fXdM. 10 CHAPTER 0. MEASURES, INTEGRATION, CONVERGENCE Proof. We give the proof under the assumption Xn —>a.e. X. Then Zn E |Xn — X| —> 0 a.e. and Zn 3 |Xn| —|— |X| g 2Y E Z. Thus Z — Zn 2 0 and by Fatou’s lemma /Zd,u : — Zn)d,u g li_m (Z — Zn)d,u : /Zd,u —E/an,u, and this implies E/anfi/pfn—qugo. Thus |/Xt—/XW=t/ua—Xymbs/mg—quso Definition 2.6 Let X be a finite meaurable function on a probability space (Q,A,P) (so that 13(9) : 1). Then X is called a random variable and .fiflDEHXEBfi:HwEQpflMEBD for all B E B is called the (induced) probability distribution of X (on R). The df associated With PX is denoted by FX and is called the df of the random variable X. Thus (R, B, PX) is a probability space. Theorem 2.4 (Theorem of the unconscious statistician.) lfg is afinite measurable function from R to R, then hémenwmn=/ R mmflflm:émmflflm Proposition 2.8 (Interchange of integral and limit or derivative.) Suppose that X(w,t) is measurable for each If 6 ((1,1)). (i) le(w,t) is a.e. continuous in t at to and |X(w,t) g Y(w) a.e. for |t—tol < 6 With Y integrable, then fX(-,t)d,u is continuous in t at 150. (ii) Suppose that %X(w,t) exists for a.e. to, all t 6 ((1,1)), and |%X(w,t)| g Y(w) integrable a.e. for all t 6 ((1,1)). Then géXmmmw=(%Mwwma Proof. (ii). By the mean value theorem X(w,t—|—h) — X(t) _ 8 ———7————EMWW3 for some t g .9 3 15+ h. Also the left side of the display converges to %X(w,t) as h —> 0 for a.e. w, and by the equality of the display and the hypothesized bound, the difference quotient on the left side of the display is bounded in absolute value by Y. Therefore g/meww):1ml{/MWHWMMM—/XWMWWfi h—>0h _ ' X(w7t+h)_X(w7t) — {fw : /%meww) Where the last equality holds by the dominated convergence theorem. 3. ABSOLUTE CONTINUITY, RADON—NIKODYM THEOREM, FUBINI’S THEOREM 11 3 Absolute Continuity, Radon-Nikodym Theorem, Fubini’s The- orem Let (Q,A,,u) be a measure space, and let X be a non—negative measurable function on 9. For A69, set 1/(A)E/Xd,u:/1AXd,u. A 9 Then 1/ is another measure on (Q,A) and 1/ is finite if and only if X is integrable (X E L1(,u)). Definition 3.1 The measure 1/ defined by 7? is said to have density X With respect to 11. Note that ,u(A) : 0 implies that 1/(A) : 0. Definition 3.2 If ,u, 1/ are any two measures on (Q,A) such that ,u(A) : 0 implies 1/(A) : 0 for any A E A, then 1/ is said to be absolutely continuous With respect to ,u, and we write 1/ << ,u. We also say that 1/ is dominated by 11. Theorem 3.1 (Radon-Nikodym theorem.) Let (Q,A,,u) be a U—finite measure space, and let 1/ be a measure on (Q,A) With 1/ << 11. Then there exists a measurable function X Z 0 such that 1/(A) : L, Xu for all A E A. The function X E Ell—:1 is unique in the sense that if Y is another such function, then Y : X a.e. With respect to 11. X is called the Radon—Nikodym derivative of 1/ With respect to ,u. Proof. See Billingsley (1986), page 376. Corollary 1 (Change of Variable Theorem.) Suppose that 1/, ,u are U—finite measures defined on a measure space (Q,A) With 1/ << 11, and suppose that Z is a measurable function such that f Zdz/ is well—defined. Then for all A E A, Proof. If Z : 113; then d d /1Bd1/:1/(AflB):/ —Vd,u:/ lB—Vdu. A AnB am A dM (ii) If Z : 2,114,, then /Zd1/ : Za/lAidz/ A 12 CHAPTER 0. MEASURES, INTEGRATION, CONVERGENCE (iii) If Z Z 07 let Zn 2 0 be simple functions / Z. Then / Zdz/ : lim anI/ by the monotone convergence thm. A d lim/Znédu by part (ii) : Z—du by the monotone convergence thm. (iv) If Z is measurable, Z : Z+ — Z‘ Where one of Z‘l’7 Z‘ is 1/—integrable7 then /Zdz/ : Z"dz/— Z_d1/ A A A d1/ d1/ : Z"—d — Z —d b iii A d” M A d” M y( ) : Zd—Vd A dM Example 3.1 Let (SLA, P) be a probability space; often this Will be (1271,1871, P). Often in statis— tics we suppose that P has a density f With respect to a U—finite measure ,u on (SLA) so that HAP/Ami for AEA. If ,u is Lebesgue measure on R”7 then f is the density function. If ,u is counting measure on a countable set7 then f is the frequency function or mass function. Proposition 3.1 (Scheffé’s theorem.) Suppose that l/n(A) : fA fndn7 that l/(A) : fA fdu Where fn are densities and l/n(9) : l/(Q) < 00 for all n, and that fn —> f a.e. M. Then sup WA) — M)! = lfn — fl a 0. AeA Proof. For A E A, WA) — um» | [4m — new Alfn—fldMS/Qlfn—flduv | /\ and this implies that sup WA) — V(A)| : /Q lfn — fldu- AeA Let gn E f — Now g; —> 0 a.e. ,u, and g; g f Which is integrable. Thus by the dominated convergence theorem ff]ng —> 0. But 0:/gndu=4(f—fn)du=é(gi—g;)dua 3. ABSOLUTE CONTINUITY, RADON—NIKODYM THEOREM, FUBINI’S THEOREM 13 so f5]ng : fggdu, and hence /lgnldu=/9$dM+/g;du=2/g$dM—>07 proving the claimed convergence. To prove that equality holds as claimed in the statement of the proposition, note that for the event B E — fn Z 0] we have 31g)“ IMA) — V(A)| Z IVn(B) — V(B)| =| [f_fn20](fn — f)dHl = /[g;20]g$du = /9$du = é/Ifn — fldu- But on the other hand hMm—WMI=|Ahw—Aflm =Ijgf—nwm = I AnB(f—-fiadu—+ AOBgf-—jtynd | /\ /g$dm sumacn—mAns/mwu=§/Ut—flwu AeA SO Now supppose that (X,X,,u) and (Y,)},V) are two U—finite measure spaces. If A 6 X7 B E )1, a measurable rectangle is a set of the form A X B C X X Y. Let X X 312 0({A X B : A E X,B E Define ameasure 7r on (XX Y,X>< )1) by mAxm=uwww> for measurable rectangles A X B. Theorem 3.2 (Fubini - Tonelli theorem.) Suppose that f : X X Y —> R is X X y—measurable and f 2 0. Then / f(x,y)dz/(y) is X — measurable , Y / f(9073/)d,u(96) is )1 — measurable , X and (u Afljflwwflaw=é{éflawWWfiwmfl=4{éfl%wWWfiWW) If f E L1(7r) (so [Xxy |f|d7r < oo)7 then (1) holds. ...
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Ch0 - Chapter 0 Measures, Integration, Convergence 1....

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