Real Analysis J. Yeh

Theory of Measure and Integration J

Yeh

REAL ANALYSIS Theory of Measure and Integration 2nd Edition

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2nd Edition

REAL ANALYSIS Theory of Measure and Integration

J Yeh University of California, Irvine

\jJ5 World Scientific NEW JERSEY • LONDON

• SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Yeh, J. Real analysis: theory of measure and integration / J. Yeh. -- 2nd ed. p. cm. Rev. ed. of: Lectures on real analysis. 1st ed. c2000. ISBN 981-256-653-8 (alk. paper) - ISBN 981-256-654-6 (pbk.: alk. paper) 1. Measure theory. 2. Lebesgue integral. 3. Integrals, Generalized. 4. Mathematical analysis. 5. Lp spaces. I. Yeh, J. Lectures on real analysis. II. Title. QA312.Y44 2006 515'.42-dc22 2006045274

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Printed in Singapore by Mainland Press

To Betty

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Contents Preface to the First Edition

xiii

Preface to the Second Edition

xvii

Notations

xix

1 Measure Spaces §0 Introduction §1 Measure on a a -algebra of Sets [I] a -algebra of Sets [II] Limits of Sequences of Sets [III] Generation of a -algebras [IV] Borel CT-algebras [V] Measure on a £ [ •

The set function JX* thus defined on ^3(R) is nonnegative extended real-valued, that is, 0 < ji* (E) < oo for every E € ^J(R), with n*(0) = 0; monotone in the sense that M*(E) < li* (F) for any £ , F e qj(R) such that E c F; and ti*(l) = 1(1) for every open interval / in R so that /x* is an extension of the notion of length to an arbitrary subset of R. The set function /x* also has the property that (2)

M*(£i U E2) < ti*(Ex) + tx*(E2)

for any two sets E\,E2 e $P(R). We call this property the subadditivity of /a* on $P(R). We say that a set function v on a collection £ of subsets of R is additive on £ if we have v(E\ U E2) = v(E\) + v(E2) whenever E\, E2 e £, E\ n E2 = 0 and Ei U £2 e £• Our set function ii* is not additive on ^J(R), that is, there exist subsets E\ and E2 of R which are disjoint but not separated enough, as far as /x* is concerned, to have {i*(E\ U E2) = H*(E\) + (i*(E2). Examples of such sets are constructed in §3 and §4. Let us show that it is possible to restrict /x* to a subcollection of ^P(R) so that 11* is additive on the subcollection. Let E e $J(R) be arbitrarily chosen. Then for every A e ^J(R), ADE and A DEC, where Ec is the complement of E, are two disjoint members of 93(R) whose union is A. We say that the set E satisfies the /x*-measurability condition and E is a /immeasurable set if (3)

/x*(A) = n*(AtlE)

+ n*(AnEc) 1

forevery A e «p(R).

2

CHAPTER 1 Measure Spaces

It is clear that if E satisfies condition (3), then so does Ec. Note also that 0 and R are two examples of members of Vp(R) satisfying condition (3). Now let 9Jt(ix*) be the subcollection of ^J(R) consisting of all ^-measurable sets in ^3(R). Let us show that 9Jl(fx*) is closed under unions. LetEi, E 2 € SPt(/x*). Then we have/z*( A) = fi*(AC\Ei) + n*(AC\Ecx) for an arbitrary A e A n B G 21,

(4) A i , . . . ,A„ G 21 =*• f l L i A* € 21, (5) A, S

G

21 =>• A \ S G 21.

Proof. (1) follows from 1° and 2° of Definition 1.1. (2) is by repeated application of 3°. Since A n B = (Ac U Bc)c, (3) follows from 2° and 3°. (4) is by repeated application of (3). For (5) note that A \ B = A D Bc G 21 by 2° and (3). • Definition 1.3. An algebra 21 of subsets of a set X is called a a-algebra (or a a-field) if it satisfies the additional condition: 4° (A„ : n e N) C 21 =• U„eN A « £

a

-

Note that applying condition 4° to the sequence (A, B, 0, 0, . . . ) , we obtain condition 3° in Definition 1.1. Thus 3° is implied by 4°. Observe also that if an algebra 2t is a finite collection, then it is a a-algebra. This follows from the fact that when 2! is a finite collection then a countable union of members of 21 is actually a finite union of members of 21 and this finite union is a member of 21 by (2) of Lemma 1.2. Lemma 1.4. //'21 is a a-algebra of subsets of a set X, then (6) (An : n e N) C 21 =• f\eN

An

G

2t.

Proof. Note that H n e N An = ( U e N AnYThus by 2°, we have (|J n e N Acn)c e 21. •

By 2°, Ac„ G 21 and by 4°, U „ e N K

e 21.

4

CHAPTER 1 Measure Spaces

Notations. For an arbitrary set X, let Vp(X) be the collection of all subsets of X. Thus A e %$(X) is equivalent to A C X. Example 1. For an arbitrary set X, %S(X) satisfies conditions 1° - 3° of Definition 1.1 and condition 4° of Definition 1.3 and therefore a a-algebra of subsets of X. It is the greatest a-algebra of subsets of X in the sense that if 21 is a a-algebra of subsets of X and if *P(X) C 21 then 21 = 0°

n->oo

Proof. 1. Let x e X. If x e A„ for all but finitely many n e N , then there exists no e N such that x e Ak for all k > no. Then x e C\k>no Ac c UneN fltsn Ak = liminf A„. Conversely if x e liminf A„ = UneN C\k>n Ajfc. then x € ( \ > n 0 A * f ° r some no e N and thus x e Ak for all A: > no, that is, x e A„ for all but finitely many n e N . This proves (1). 2. If x e A„ for infinitely many n e N , then for every n e N we have x e \Jk>n Ak and thus x e f|„eN U*> n A * = l i m SUP A « • Conversely if x e lim sup A„ = f]neN \Jk>n Ak, n—>oo

n—>-oo

then JC e \Jk>n Ak for every n e N . Thus for every n € N, x e Ak for some k > n. This shows that x e A , for infinitely many n e N . This proves (2). 3. (1) and (2) imply (3). I Definition 1.8. Let (A„ : n e N) be an arbitrary sequence of subsets of a set X. If lim inf A„ = lim sup A„, then we say that the sequence converges and define lim An by n->cx>

n->-oo

n-*oo

setting lim A„ = liminf A„ = lim sup A„. //"liminf An ^ lim sup A„, then lim A„ n-*oo

n->oo

„_>„-oo

n_+00

„->.oo

does nof erisf. Note that this definition of lim A„ contains the definition of lim A„ for monotone n->oo

n—yoo

sequences in Definition 1.5 as particular cases and thus the two definitions are consistent. Indeed if A„ f then f \ > „ At = A, for every n e N and U „ e N f \ > „ Ak = L L N A « and therefore liminf A„ = UneN A «- O" m e other hand, \Jk>n Ak =\jkeN Ak for every n e N and f)„€N Ut>« At = Uoo

Example. Let X = R and let a sequence (A„ : n e N) of subsets of R be defined by Ai = [0, 1], A3 = [0, ±] , A5 = [0, ± ] , . . . , and A2 = [0, 2], A4 = [0, 4], A6 = [0, 6], — Then lim inf A„ = {x e X : x e A„ for all but finitely many n e N ) = {0} and lim sup A„ = {x € X : x e A„ for infinitely many n € N} = [0, oo). Thus lim A„ does n->oo

n-»oo

not exist. The subsequence (A„k : k e N) = (A\, AT, , As,...) is a decreasing sequence with lim A„k = (0} and the subsequence (A„t : k e N) = (A2, A4, A 6 , . . . ) is an k—»oo

increasing sequence with lim A„t = [0, 00). k—»-oo

Theorem 1.9. Lc? 21 be a a-algebra of subsets of a set X. For every sequence (A„ : n e N) in 21, fne fwo sets lim inf A„ ana" lim supA„ are in 21. So is lim A„ i/zf exwri. n—roo

n—•oo

n-^-oo

Proof. For every n € N, f)t> n At e 21 by Lemma 1.4. Then U „ £ N flt> n At e 21 by 4° of Definition 1.3. This shows that liminf A„ € 21. Similarly I J.. Ak e 21 by 4° of Definition 1.3. Then DneN U*> n At

e

21 by Lemma 1.4. Thus lim sup A„ e 21. If n->oo

lim A„ exists, then lim A„ = lim inf An e 21. • «—>oo

n—>-oo

n—>-oo

[III] Generation of cr-algebras Let A be an arbitrary set. If we select a set Ea corresponding to each a e A, then we call {£0; : a € A] a collection of sets indexed by A. Usual examples of indexing set A are for instance N = (1,2, 3 , . . . } , Z = {0, 1, - 1 , 2, - 2 , . . . }, and Z+ = {0, 1 , 2 , . . . } . An arbitrary set A can serve as an indexing set. Lemma 1.10. Let {2l a : a G A] be a collection of a -algebras of subsets of a set X where A is an arbitrary indexing set. Then C\a€A 2l a is a a-algebra of subsets ofX. Similarly if {2la : a € A] is an arbitrary collection of algebras of subsets of X, then f^\a€A 2l a is an algebra of subsets ofX. Proof. Let {2l a : a e A} be an arbitrary collection of o-algebras of subsets of X. Then DaeA %la is a collection of subsets of X. To show that it is a cr-algebra we verify 1°, 2°, and 3° in Definition 1.1 and 4° in Definition 1.3. Now X e 2l a for every a e A so that X e f]a€A 2. Then (Gn : n e N) is a disjoint sequence in 21 such that UneN *^" = UneN Fn = D as in the Proof of Lemma 1.21, Ai(Gi) = /x(Fi) < oo and /x(G„) = /x(F„ \ (J!fe=i F 0 < M(^») < oo for « > 2. This proves (a). 2. Let (X, 21, /x) be aCT-finitemeasure space. Then there exists a sequence (F„ : n e N) in 21 such that | J n £ N E„ = X and /U,(F„) < oo for every n € N. Let D € 21. For each w e N, let D„ = D n En. Then (£>„ : n e N) is a sequence in 2t such that |J„ e N Dn = D and ix(Dn) < fi(En) < oo for every n e N. Thus D is a er-finite set. This proves (b). I Definition 1.32. Given a measure /x on a a-algebra 21 of subsets of a set X. A subset E of X is called a null set with respect to the measure \x if E e 21 and ix(E) = 0. In this case we say also that E is a null set in the measure space (X, 21, it). (Note that 0 is a null set in any measure space but a null set in a measure space need not be 0.) Observation 1.33. A countable union of null sets in a measure space is a null set of the measure space. Proof. Let (En : n G N) be a sequence of null sets in a measure space (X, 21, fi). Let E = UneN ^«- Since 21 is closed under countable unions, we have E e 21. By the countable subadditivity of p. on 21, we have p-(E) < X!neN ViEn) = 0- Thus /x(F) = 0. This shows that E is a null set in (X, 21, //,). • Definition 1.34. Given a measure fiona a-algebra 21 of subsets of a set X. We say that the a-algebra 21 is complete with respect to the measure \JL if an arbitrary subset EQ of a null set

§1 Measure on a cr-algebra of Sets

19

E with respect to p is a member o/2l (andconsequently has p(Eo) = 0 by the monotonicity of p). When 21 is complete with respect to p, we say that (X, 21, p) is a complete measure space. Example. Let X = {a, b, c}. Then 21 = {0, {a}, {b, c), X\ is a cr-algebra of subsets of X. If we define a set function p on 21 by setting p(&) = 0, p({a}) = 1, p({b, c}) = 0, and p(X) = 1, then p is a measure on 21. The set {b, c] is a null set in the measure space (X, 21, p), but its subset {b} is not a member of 21. Therefore (X, 21, /it) is not a complete measure space. Definition 1.35. (a) Given a measurable space (X, 21). An SH-measurable set E is called an atom of the measurable space if® and E are the only %L-measurable subsets of E. (b) Given a measure space (X, 21, p). An ^-measurable set E is called an atom of the measure space if it satisfies the following conditions : 1°

p(E)

2°

E0 C E, E0 6 21 => p(E0) = 0 or p(E0) = p(E).

> 0,

Observe that if E is an atom of (X, 2t) and p(E) > 0, then E is an atom of (X, 21, p). Example. In a measurable space (X, 21) where X = {a, b, c) and 21 = {0, (a), {b, c}, X], if we define a set function p on 21 by setting /x(0) = 0, /•'({a}) = 1, p({b, c}) = 2, and p(X) = 3, then p is a measure on 21. The set {&, c} is an atom of the measure space (X,St,At).

[VIII] Measurable Mapping Let / be a mapping of a subset D of a set X into a set y. We write S ) ( / ) and £H(/) for the domain of definition and the range of / respectively. Thus 2>(/) =

DCX,

(/)} C Y. For the image of £ ) ( / ) by / we have / ( © ( / ) ) = JH(/). For an arbitrary subset E of y we define the preimage of E under the mapping / by f~\E)

:= {* s X : f(x) e E) = {x e © ( / ) : fix)

e E}.

Note that E is an arbitrary subset of Y and need not be a subset of 9 t ( / ) . Indeed E may be disjoint from 9 t ( / ) , in which case f~l(E) = 0. In general we have / ( / " ' ( £ ) ) C £ . For an arbitrary collection (/) \ / - * ( £ ) ,

Proposition 1.37. Given sets XandY. Let f be a mapping with 55(f) C XandV>\{f) C F. 7^23 is a a-algebra of subsets ofY then / _ 1 ( 2 3 ) w a a-algebra of subsets of the sefD(f). In particular, ifZ>(f) = X then / - 1 ( 2 3 ) is a a-algebra of subsets of the set X. Proof. Let 23 be a cr-algebra of subsets of the set Y. To show that / _ 1 ( 2 3 ) is a cralgebra of subsets of the set 2>(/) we show that £>(/) s / _ 1 ( 2 5 ) ; if A e / _ 1 ( 2 3 ) then 2 ? ( / ) \ A e / - 1 ( 2 3 ) ; and for any sequence (A„ : n 6 N) in / _ 1 ( 2 3 ) we have 1. By (1) of Observation 1.36, we have £>(/) = f~x(Y) e / _ 1 ( 2 3 ) since Y e 93. 2. Let A e / - 1 ( 3 3 ) . Then A = / _ 1 ( £ ) for some B e 23. Since Bc e 23 we have l c f~ (B ) e / _ 1 ( 2 3 ) - On the other hand by (2) of Observation 1.36, we have f~l(Bc) = » ( / ) \ / " ' ( B ) = S ) ( / ) \ A. Thus £>(/) \ A e / " ' ( S B ) . 3. Let (A„ : n s N) be a sequence in / _ 1 (23). Then An = f~1(Bn) for some B„ e 23 for each n e N. Then by (3) of Observation 1.36, we have

U A" = U f'l(Bn) = / " ' ( U B n ) € neN

since \Jn€n Bn e 23.

neN

f~l(*^

neN

•

Definition 1.38. Given two measurable spaces (X, 21) and (F, 23). Let f bea mapping with 2>(/) C X and 9 t ( / ) C Y. We say that f is a 21/23 -measurable mapping iff'1 (B) e VL for every B e 23, r/zar is, / _ 1 ( 2 3 ) c 21. According to Proposition 1.37 for an arbitrary mapping / of £>(/) C X into y, / - 1 (23) is a cr -algebra of subsets of the set 2) ( / ) . 21/23 -measurability of the mapping / requires that the cr-algebra f"1 (23) of subsets ofD(f) be a subcollection of the cr-algebra 21 of subsets of X. Note also that the 2l/23-measurability of / implies that £>(/) = f~l(Y) e 21. Observation 1.39. Given two measurable spaces (X, 21) and (Y, 23). Let / be a 21/23measurable mapping. (a) If 2li is a cr-algebra of subsets of X such that 2ti D 21, then / is 2li/23-measurable. (b) If 23o is a cr-algebra of subsets of Y such that 23o C 23, then / is 2t/23o-measurable. Proof, (a) follows from / _ 1 ( 2 3 ) C 21 C 2ti and (b) from / " 1 ( 2 3 0 ) C / " ' ( » ) C 21. I Composition of two measurable mappings is a measurable mapping provided that the two measurable mappings form a chain. To be precise, we have the following:

§1 Measure on a a -algebra of Sets

21

Theorem 1.40. (Chain Rule for Measurable Mappings) Given measurable spaces (X, 21), (Y, 23), and (Z, €). Let f be a mapping with £>(/) C X, 9 t ( / ) C K, g fee a mapping with 2>(g) C F, IH(g) C Z SMC/J f/iaf $H(/) C 2>(g) so r/zar ffte composite mapping g o / w defined with 1>(g o f) C X and 9\(g o / ) c Z. /f / is 21 /'33-measurable and g is *B/-oo

= 1A on X then lim A„ = A. n-*oo

Prob. 1.5. Let 21 be a a-algebra of subsets of a set X and let 7 be an arbitrary subset of X. Let 23 = {A n Y : A e 21}. Show that 23 is a a -algebra of subsets of Y. Prob. 1.6. Let 21 be a collection of subsets of a set X with the following properties: 1°. X e 2 l , 2°. A, fi € 21 => A \ B = A n S c e 21. Show that 21 is an algebra of subsets of the set X. Prob. 1.7. Let 21 be an algebra of subsets of a set X. Suppose 21 has the property that for every increasing sequence (A„ : n e N) in 21, we have UneN -A„ e 2 1 . Show that 21 is a CT-algebra of subsets of the set X. (Hint: For an arbitrary sequence (Bn : n e N) in 21, we have (J«eN ^« = UneN ^« where (A„ : n € N) is an increasing sequence in 21 defined by A„ = U*=i ^* f ° r w e ^-) Prob. 1.8. (a) Show that if (2l„ : n e N) is an increasing sequence of algebras of subsets of a set X, then UneN ^t« is an algebra of subsets of X. (b) Show by example that even if 2l„ in (a) is a a -algebra for every n e N, the union still may not be a a -algebra. Prob. 1.9. Let (X, 21) be a measurable space and let (En : n e N) be an increasing sequence in 21 such that UneN En = X(a) Let 2t„ = 21 n En, that is, 2l„ = {A n E„ : A e 2l}. Show that 2l„ is a a-algebra of subsets of En for each n e N. (b) Does U„eN ^ = 21 hold? Prob. 1.10. Let X be an arbitrary infinite set. We say that a subset A of X is co-finite if Ac is a finite set. Let 21 consist of all the finite and the co-finite subsets of a set X. (a) Show that 21 is an algebra of subsets of X. (b) Show that 21 is a cr-algebra if and only if X is a finite set. Prob. 1.11. Let X be an arbitrary uncountable set. We say that a subset A of X is cocountable if Ac is a countable set. Let 21 consist of all the countable and the co-countable subsets of a set X. Show that 21 is a a -algebra. (This offers an example where an uncountable union of members of a er-algebra is not a member of the a-algebra. Indeed, let X be an uncountable set and let A be a subset of

24

CHAPTER 1 Measure Spaces

X such that neither A nor Ac is a countable set so that A, Ac g 21. Let A be given as A = \xy e X : y e r] where T in an uncountable set. Then {xy} e 21 for every y e V, but{Jyer{xr) = A