Measure theory and integration (second edition)

Chapter 2 Measure on the Real Line2.3 Regularity Chapter 3 Integration of Functions of a Real Variable... 6 ContentsChapter 5 Abstract Measure Spaces Chapter 6 Inequalities and the L" Sp

MEASURE THEORY AND INTEGRATION

"Talking of education, people have now a-days" (said he) "got a strangeopinion that every thing should be taught by lectures Now, I cannot seethat lectures can do so much good as reading the books from which thelectures are taken I know nothing that can be best taught by lectures,except where experiments are to be shewn You may teach chymestry bylectures —Youmight teach making of shoes by lectures!"

James Bosweil: Life of Samuel Johnson, 1766

ABOUT OUR AUTHORGearoid de Barra was born in the city of Galway, West Ireland and moved

as a young boy to Dublin where he spent his schooldays He then studiedmathematics at University College Dublin, National University of Irelandwhere he gained his BSc Moving to England, he graduated from the University of London with a PhD for research on the convergence ofrandom variables, an area of application of some of the material covered inthis book He then transferred to Hull University in Yorkshire for a teachingappointment; and afterwards spent two summers in 1975 and 1988 inAustralia, teaching and researching at the University of New South Wales.More recently, he became Senior Lecturer at the Royal Holloway College,University of London, to continue teaching and research related to aspects

of operator theory and measure theory involving ideas from the material inthis book

He has enjoyed teaching university mathematics at all undergraduate andpostgraduate levels, including many courses on measure theory and its applications to functional analysis, from which source this book hasdeveloped The first edition was the standard text in the departments ofmathematics at both Cardiff University and Royal Holloway College, andhas attracted attention in Canada and Scandinavia It was also translated intoItalian as Teoria del/a Misura è deliintegrazione

Measure Theory and Integration

Published by Woodhead Publishing limited.

80 1 ugh Street Cambridge CR22 31 Ii

oodheadpuhl ish ing.eom

Woodhead Publishing 1518 Walnut Street Suite 1100 Philadelphia.

PA 19102-3406 USA

Woodhead Publishing India Private Limited (1-2 Vardaan house 7/28 Ansari Road.

Daryaganj Ness 1)elhi — 110002 India

First published 1981

Updated edition published h' I Publishing Limited 2003

Reprinted by Woodhead Publishing limited 2011

(I de Rarra 2003

'I he author has asserted his moral rights

[his book eontaiils information obtained from authentic and highly regarded sources Reprinted nìaterial is quoted permission and sources arc indicated Reasonable efforts have been made to publish reliable data and information but the author and the publisher cannot assume responsibility for the validity of all materials Neither the author nor the publisher, nor anyone else associated ith this publication, shall he liable for an\ loss, damage or liability directly or indirectly caused or alleged to he caused by this hook

Neither this hook nor any part may he reproduced or transmitted in any form or by' any means electronic or mechanical including photocopy ing microt'ilming and recording or by' an) inforniation storage or retrieval sstcm ithout permission in from Woodhead Publishing limited.

I he consent of Woodhead Publishing Limited does not c\tend to copying for general distribution, for promotion for creating works or for resale Specific permission must he obtained in writing froni Woodhead Publishing limited for such copying.

1 radcmark notice: Product or corporate names may he trademarks or registered trademarks and arc used only for identification and explanation ithout intent to

infringe.

Rritish Library' Cataloguing in Publication l)ata

A catalogue record for this hook is from the Rritish I ibran

ISBN 978-1-904275-04-6

Printed by I ightning Source.

Chapter 2 Measure on the Real Line

2.3 Regularity

Chapter 3 Integration of Functions of a Real Variable

6 Contents

Chapter 5 Abstract Measure Spaces

Chapter 6 Inequalities and the L" Spaces

Chapter 8 Signed Measures and their Derivatives

8.4 Some Applications of the Radon-Nikodym Theorem 142

Chapter 9 Lebesgue-Stieltjes Integration

Contents 7

Chapter 10 Measure and Integration in a Product Space

Hints and Answers to Exercises

Preface to First Edition

This book has a dual purpose, being designed for a University level course onmeasure and integration, and also for use as a reference by those more interested

in the manipulation of sums and integrals than in the proof of the mathematicsinvolved Because it is a textbook there are few references to the origins of thesubject, which lie in analysis, geometry and probability The only prerequisite is

a first course in analysis and what little topology is required has been developedwithin the text Apart from the central importance of the material in puremathematics, there are many uses in different branches of applied mathematicsand probability

In this book I have chosen to approach integration via measure, rather thanthe other way round, because in teaching the subject I have found that in thisway the ideas are easier for the student to grasp and appear more concrete.Indeed, the theory is set out in some detail in Chapters 2 and 3 for the case of thereal line in a manner which generalizes easily Then, in Chapter 5, the results forgeneral measure spaces are obtained, often without any new proof The essentialL" results are obtained in Chapter 6; this material can be taken immediately afterChapters 2 and 3 if the space involved is assumed to be the real line, and themeasure Lebesgue measure

In keeping with the role of the book as a first text on the subject, the proofsare set out in considerable detail This may make some of the proofs longer thanthey might be; but in fact very few of the proofs present any real difficulty.Nevertheless the essentials of the subject are a knowledge of the basic resultsand an ability to apply them So at a first reading proofs may, perhaps, beskipped After reading the statements of the results of the theorems and thenumerous worked examples the reader should be able to try the exercises Over

300 of these are provided and they are an integral part of the book Fairlydetailed solutions are provided at the end of the book, to be looked at as a lastresort

Different combinations of the chapters can be read depending on thestudent's interests and needs Chapter 1 is introductory and parts of it can beread in detail according as the definitions, etc., are used later Then Chapters 2and 3 provide a basic course in Lebesgue measure and integration Then Chapter

4 gives essential results on differentiation and functions of bounded variation, allfor functions on the real line Chapters 1, 2, 3, 5, 6 take the reader as far asgeneral measure spaces and the L" results Altematively, Chapters 1, 2, 3, 5, 7

9

10 Preface

introduce the reader to convergence in measure and almost uniformconvergence To get to the Radon-Nikodym results and related material thereader needs Chapters 1, 2, 3, 5, 6, 8 For a course with the emphasis ondifferentiation and Lebesgue-Stieltjes integrals one reads Chapters 1, 2, 3, 4, 5,

8, 9 Finally, to get to measure and integration on product spaces the appropriateroute is Chapters 1, 2, 3, 5, 6, 10 Some sections can be omitted at a firstreading, for example: Section 2.6 on Hausdorff measures; Section 4.6 on theLebesgue set; Sections 8.5 and 9.6 on Riesz Representation Theorems andSection 9.2 on Hausdorff measures

Much of the material in the book has been used in courses on measure theory

at Royal Holloway College (University of London) This book has developedout of its predecessor introduction to Measure Theory by the same author(1974), and has now been rewritten in a considerably extended, revised andupdated form There are numerous proofs and a reorganization of structure Theimportant new material now added includes Hausdorif measures in Chapters 2and 9 and the Riesz Representation Theorems in Chapters 8 and 9

Ode BarraPreface to Second Edition

The material in this book covers several aspects of classical analysis includingmeasure, integration with respect to a measure and differentiation These topicslead on to many branches of modem mathematics So some notes have beenadded which indicate some of the directions in which the material leads Theseare less formal than the main text and contain some references for furtherreading

Ode BarraRoyal Holloway, University of London

January 2003

x CA: x is a member of the set A.

A c B, (A 2 B): set A is included in (includes) the set B

A CB: set A is a proper subset of the set B

Ex: P(x)I the set of those x with property P

CA: the complement of A

0: the empty set

U, fl: union,intersection (of sets)

A —B the set of elements of A not in B

A A B =(A —B)U(B —A): the symmetric difference of the sets A, B

Z: integers (positive or negative)

N: positive integers

Q: rationals

R: real numbers

?(A): the power set of A, i.e the set of subsets of A

A X B: the Cartesian product of the sets, A, B

[xl: the equivalence class containing x (Chapter 1), or, in Chapter 2, etc., theclosed interval consisting of the real number x

Ix, the metric space consisting of the space X with metric p

A: the closure of the set A

G5 set: one which is a countable intersection of open sets

set: one which is a countable union of closed sets

11

12 Notation

inf A, sup A: infirnum and supremum of the set A

Urn sup x,,,liin upper and lower limits of the sequence (x,,)

x(a—),x(a+): left-hand, right-hand limits of x at a So x(a+) is the functionwhose value at a is lim -* a, > a Similarly flx+),ftx—), etc.x,, = o(n"): ,f1'x,, 0.

x,, =O(n"): x,, is bounded

characteristic function of the set A (=I on A, =0on CA)

Card A: the cardinality of A

the cardinality of N

(2(a): (in Theorem 9, Chapter 1) the equivalence class containing a

Cantor-like sets

'1,1 etc.: the 'removed intervals';

the 'residual intervals', for the Cantor-like sets

N(x,e): the set [z': It—xi<e]

L: the Lebesgue function

m*: Lebesgue outer measure

m: Lebesgue measure

A+x= [y+x:yEA1.

1(1): the length of the interval I

a-algebra (usually 8): a class closed under countable unions and complementsand containing the whole space

Intervals: of the form [a, b) unless stated otherwise

the a-algebra of Lebesgue measurable sets

a.e almost everywhere; i.e except on a set of zero measure

a-algebra of Borel sets

r =max(f, =—min(f,0)

esssupf= thf[a:fCaa.e.J essinff= a.e.j

lim A,, lim sup A,, lim infA1: the limit, upper limit, lower limit of the sequence

of sets IA,)

r'(A)= (x:/(x)EEA].

r= [x—y:x,yETJ (Chapter2).

d(A,B)' inf Fix —yI:x EA,y EBJ.

h: a Hausdorff measure function

H;,6: the 'approximating measure' to l-lausdorff measure

H;: Hausdorff outer measure when h(t) =tP.

modulusof continuity of the functionf

H(A): Hausdorff measure corresponding to the Hausdorff measure function h.ffdx: integral (over the whole line) off with respect to Lebesgue measure.ffdx: integral offover the set F

0, p4.': usuallysimple functions, taking only a finite number of non-negative values.Rffdx: Riemann integral off

ifi: absolute value of the real (or in Chapter 10, complex) functionf

Notation 13

logx the natural logarithm of x

5D' SD: upper and lower Riemann sums given by the dissection D

L(a, b): functions integrable on (a, b)

fh(X)=f(X + h)

upper and lower right derivates

IX, 11: upper and lower left derivates

Pf[a, b], N1[a, b], T1[a, b]: positive, negative and total variations of f over[a,b].

p (or p4, n (or n/I, t (or t1) the corresponding sums for a partition

BV[a, hi: set of functions of bounded (total) variation on [a, bJ

1(w): length of the polynomial ir

81(x): the lump' offatx.

ftc, 6) =(f(d)—f(c))/(d —c), wheref is a function of a single variable

f'(x) =df/dx

F: Conventionally, the indefInite integral off

'1? : ring of sets (closed under unions and differences)

S (R): a-ring generated by R

2KQR): hereditary a-ring generated by 'It

a-finite measure: one for which the space is a countable union of sets of finitemeasure

pt: any outer measure, or the outer measure defined by p

8 *: class of p*measurable sets

p: measure obtained by restricting /1* to S *, also the completion of the measurep

g: a-ringobtained on completing measure p on S

(1, 8 j: measurable space

11,8 ,pI: measure space

f= limf,,: pointwise liniit,f(x) = eachx

ffdp: integral off with respect to p

ffdp: integral off over the set E

L(X, p): class of functions integrable with respect to p

or L"(p): the class of measurable functions with f f1P dp Cootions equal a.e being identified

func-if I,, =(fIfI" dp)1'T', the 9-norm off

'4i of: composite function, (ji of)(x)=

/',, -÷ fa.u.: f, -+f almost uniformly (uniform convergence with an exceptionalset of arbitrarily small measure)

pip: measures v, p mutually singular, i.e i4A) =p(C,4)=0for some measurableA

v = — ii: Jordan decomposition of the signed measure v.

total variation of the signed measure v

p'<p:v is absolutely continuous with respect top, i.e p(E) = 0 v(E)=0.

14 Notation

dv/dp: Radon-Nikodym derivative of v with respect to p

[ji]:(Chapter8) indicates that an identity holds except on a set of zero p-measure,(or zero Ipi-measure for a signed measure p)

Ox II: norm of the vector x

=sup (If(x)l) =supremumnorm off

P,: Lebesgue-Stieltjes measure, with g a monotone increasing left-continuousfunction and Pg([a, b)) =g(b) —g(a)

$'g: thePg-measurable sets

I I dPg or f f dg: integral of I with respect to the Lebesgue-Stieltjes measurederived fromg Also ffdg wheng EBV[a, bJ (Definition 4, Chapter 9).pr1: the measure such that pf' (E) = (E))

C(T): the set of functions continuous on the interval I with supremum norm

&: elementary sets, i.e union of a finite number of disjoint measurable rectangles

it, monotone classes (Chapter 10)

IxX Y, S X product of measurable spaces

U': (x,y)EE].

E":y-sectionofE= [x:(x,y)EE].

fl: class of sets, depending on context (Chapter 10)

pX v: the product measure (So (p X PXA X B) = p(A)v(B))

CHAPTER 1

Preliminaries

In the chapter we collect for reference the various mathematical tools needed inlater chapters As the reader is presumed to be familiar with the content of afirst course on real analysis, we are concerned not with setting up the theoryfrom stated but with giving definitions and stating results and theoremsabout sets, sequences and functions which serve to fIx the notation and to make

it clear in what form elementary results will be used Proofs are provided for theless familiar results in section 1.7 we describe in some detail the special sets ofCantor These sets and the functions associated with them will be referred tofrequently in later chapters

The standard abbreviations: iff 'if and only if, 3 'there exists', v 'given any','implies', will be used as required The end of a proof is indicated by thesymbol 0

1.1 SET ThEORY

Whenever we use set theoretic operations we assume that there is a universal set,

X say, which contains all the sets being dealt with, and which should be clearfrom the context The empty set is denoted by 0;xE A means that the element

x belongs to the set A By A B we mean that x E A x E B; A C B is strictinclusion, that is,A B and there existsx withx EBandxnotmA

We denote by [x:F(x)] the set of points or elements x of X with the property

P The CA of A is the set of points x of X not belonging to A; CAobviously depends on the sets X implied by the content — in fact X is usuallythe set of real numbers, except in Chapter 10 We will denote the union of twosets by A U B or of a collection of sets by U where I denotes some indexset, or by :F(a)J — the union of all sets such that a has the property

P Sinularly for intersections A fl B, etc Then unions and intersections are

enceA —B =Á ñ CB;A —B)U(B—A)isthesymmetricdifference

of A and B, some properties of which are listed in Example 1 The Cartesianproduct XX Y is the set of ordered pairs [(x,y): x EX,y E YJ We wifi denote

15

16 Preliminaries [Ot I

the real numbers by R, the integers by N, the set of allintegersby Z = [0,± 1, ±

2, ], the set of rationals by 0, and n-dimensional Eucidean space by so

that is the set of n-tuples (x1, considered as a vector space with theusual inner product Notations for intervals are [a,b] = [x: a [a,b) [x: <bl etc.'P(A) denotes the set of all subsets of the set A

Example 1: Show that the following set relations hold:

Solution:(i) is obvious from the symmetry of the definition

To obtain (ii) use the identity C(E F) = (CE fl CF) U (E fl F), to get

Bysymmetry this must equal the right hand-side

(lii) (E F) (G H) =((F E) G) H by (i) and (u)

=

=

=

(iv) is obvious

(v)We haveE—Fc (E—G)U(G—F)andF—Ec (F—G)U(G—E),so

taking the union gives the result

(vi) This follows from the more obvious inclusion

UE1 — U F1=Li (E1 — F1).

Example 2: LetE1 E2 ShowthatL) (E1 —E1)=E1

Solution: This is just an application of De Morgan's laws with E1 as the wholespace

PrincipleofFinite Induction LetP(n) be the proposition that the positive integer

n has the property P If P(1) holds and the truth of P(n) implies that ofF(n+1),then P(n) holds for all n E N.It is to this property of positive integers that weare appealing in our frequent 'proofs by induction' or in inductive constructions.DefinItion 1: An equivalence relation R on a set E is a subset of E X E with thefollowing properties:

SeC 1.21 Topological Ideas 17

(i)(x,x)ER foreachxEE,

(ii)(x,y)ER

(iii) if (x,y)ER and (y,z)ER then (x,z)ER

We write x "y if(xy)ER Then R partitions E into disjoint equivalence classessuch that x and y are in the same classif, andonly if, x "y For, writing [xJ =

Lz: z " xl,we have x E by (i), so x EEl =E.Also by (ii) and (iii),

for any x,y E, either [xJ = [y] or [xl [y] = O,so the sets [xJ are therequired equivalence classes

In Chapter 2 we will need the Axiom of Choice which states that if [Es: a E AJ

is a non-empty collection of non-empty disjoint subsets of a set X, then thereexists a set V ç X containing just one element from each set

(i) p(xy)= 0 if, and only if,x

(ii) p(x,y) =p(y,x),

(ill) p(x,z) <p(x,y) + p(y,z), for any x,y, z E X

The function p then defines a distance between points of X, and the pairIX,pI forms a space If we relax the condition that the distance betweendistinct points be strictly positive so that (i) reads: p(x,y) = 0if x =y,we tam a pseudo-metric We will be especially concerned with the space R and,briefly in Chapter 10, with A" But the idea of convergence in a metric space isimplied in many of the definitions of Chapters 6 and 7

A set A in a space is open if given x EA there exists e >0 suchthat [y: p(y,x) <€1 A That is: A contains an 'e-neighbourhood' of x, denotedN(x,e) So X and 0 are open and it also follows that any union of open sets isopen and that the intersection of two open sets is again open The class of opensets of X forms a topology on X We now define various other ideas which can

be derived from that of the metric on X The properties that follow inunediatelyare assumed known In the case of the real line we list various properties whichwill be needed, in Theorem 1 and in the later sections

A set A is closed if CA is open The closure A of a set A is the intersection ofall the closed sets containing A, and is closed A point x is a limit point of A ifgiven e >0, there exists y E A, y * x, with p(y,x) <e We say that A is a denseset, or A is dense in X if A = X The set A is nowhere dense if A contains nonon-empty open set, so that a nowhere dense set in A is one whose closurecontains no open interval A is said to be a perfect set if the set [x:x a limit

18 Preliminaries [Cli 1

pointof A] is A itself If the set A may bewritten as A = G1,where the sets

_

1=1

G,are open, A is said to be a G6 -set; if A UF1where the sets F1 are closed,

thenA is an F0-set Clearly the complement of a G6 -set is an Fa-set

In Chapter 10 we will refer to the relative topology on a subset A of R",namely the class of sets G of the form H flA where H is an open set in fl" Thisclass of sets forms a topology on A

We will assuim the notion of supremum (orleast upper bound) of a setA ofreal numbers, denoted usually by sup[x: x E Al or by sup[x: P(x)1, whereP isthe property satisfied by x In the cases where we use this notation the set inquestion will be non-empty For a finite set we will write x1 for the supre-mum of the relevant set Similarly we will write mf[x: x EAI for the infimum(or greatest lower bound) or mm Xjinthe finite case.

1<i<n

We will need the following important property of the real numbers

Theorem I (Heme-Borel Theorem): If A is a closed bounded set in R and A c

U Ga, where the sets Ga are open and I is some index set, then there exists a

(ii) In the notation of p.' 17,1 let be defined by p*([xl, [yJ) =p(x,y);

show that is a metric on the set of equivalence classes

2 Show that A is nowhere dense iff cAisdense

3 Show by examples that G6 and F0-sets may be open, closed or neitheropen nor closed

& Show that In a metric space each point is a closed G5 -set

S Show by examples that Theorem 1 can break down if either of the conditions

A closed or A bounded is omitted

1.3 SEQUENCES ANDLIMITS

A numerical sequence J is a function from N to R We define the upper

=inf[ sup xm: n EN] If there is no ambiguitym>n

possible we will write this as lim sup x,, Similarly: lim inf Xm: fl E

m>n

N is the lower limit of }. Iflim sup x, hin we write their common

Sec 1.31 Sequencesand Limits 19

value as lAm x,,, the limitof }.From the definitions we get easily lim sup x,, =

—un inf (—xe)

If we consider a function from A to A we get the analogous definition: theupperlimitof at a0 is lim sup= inf[ sup xa: h >0] Similarly we define

O<Ia—a01<h

urninfxa; theircommon value, if it exists, is lim xa In the more usual tional notation we write lim sup x(a) and urn inf x(a), where x(a) is avaluedfunction A property of upper and lower limits is given by the followingexample

func-Example 3: Prove that if Jim Ya exists, then urn sup (xa + ya) =timsup xa +

a-÷a0

Iijii lim inf (xa + =liminf xa + lim where all the limits are

-'a0

supposedfinite

Solution: We prove the first equation Write

urn sup(xa+ ya), 12 = urnsup xa, 13 = urn

Given 0, there exists & > 0 such that xa <12 + and <13 + when

0 < — a0I <& SO Xa + <12 +13 + 2€ in this range, and as isarbitrary

<12 + 13 Conversely: there exists &' > 0 such that xa + + and

>13 — when 0 < a —a0I <&', so in this range xa (xa + Ya) —Ya < +2€andsol2 givingtheresult

A similar result holds for sequences

We will be particularly concerned with 'one-sided' upper and lower limits,and express these in functional notation:

limsupx(t)=inf[ sup x(a—u):h>O],

0<u<hliminfx(t) =sup[ inf x(a—u):h>O].

0<u<h

Ifthese quantities are equal, we say that km x(t) exists and we write thislimit as x(cr—) Note that x(a—) need not be defined, although lim sup x(t) andtim ml x(t) always are Replacing a — u by a + u in these definitions we get

a —

Jimsupx(t), lim ml x(t) and, if it exists, x(a+)

The sequence is monotone increasing, and we write x,,t,if for each

ii N, x,, for each n; so if a sequence Is both monotone increasing andmonotone decreasing it is constant We will assume the result that if is a

monotone sequence and is bounded, then it has a limit, and we write x,, t x or

20 Preliminaries [Ot Ix,, x as appropriate; if tx,,) is monotone but not bounded, then x,, -+oo or

x,, asthe case maybe.

A sequence isa Cauchy sequence if for any positive e there exists N suchthat lx,, — I< e for n,m >N We will assume the result that a sequence con-verges if, and only if, it is a Cauchy sequence We define a Cauchy sequence ofelements of a space similarly, requiring that P(X,,, Xm) <efor n,m > N.Then the space is a complete metric space if every Cauchy sequence converges,

so that the result assumed above is that R forms a complete metric space withp(x,y)=

We use the o- and 0-notations; so that if fx,, is a sequence of real numbers,

means that is bounded For functions from R to R: f(x) = o(,g(x)) as

x -÷ a means that given e> 0, there exists > 0 such that fr(x)t < for

0 < — aI <5 ; flx) = o(g(x)) as x -÷ meansthat given e > 0, there exists

K > 0 such that lf(x)I < for x > K Similarly J1:x) = O(g(x))means thatthere exists M> 0 such that lf(x)l Mk(x)I as x -÷a or x -÷ asthe case may

givenM> 0 there exists N such that x,,1,, > M for all n,m > N

If one index, m or n, is kept fIxed, J isan ordinary 'single' sequence and

we have the usual notation of iterated limits lim lim x11,, and thu tim

Theorem2: If ix,,,,, } isincreasing with respect to n and to m, then lim x,,,,,

n/n-,exists and we have

andif any one is infinite, all three are

Proof: Write Ym= lim this is clearly defined for each m Since

for each n, we have Ym Ym+i. So 11 = limYm exists (it may =oo).

Write 1 = sup(x,,,m: n, m E N], where we may have 1 = Then it is obviousthat lim x,,1,, =1 In the case 1 <°°wewish to show that 1=11 Since x,,,,,,

1 for each n and m, it is obvious that 1 Also, given e >0 let N be such that

and as e is arbitrary, 1 , so 1 = Similarly we may show that the third limit

in (1.1) exists and equals 1 The case = is considered similarly 0

Sec 1 4J Functions and Mappings 21

In the case of a series a, we may consider the sequence } of its partialsums,

= a1, and in the case of a double series

a1 So to each property of sequences there corresponds one of

1=1 i=1

series For instance, to Theorem 2 corresponds the following result

Theorem 3: If a1,,> 0 for all i,/ E N, then a1, exists and

1,1=1 j=1 1=1 i=1 1=1

allthree sums being infinite if any one is

We will also need the following elementary properties of series

Theorem 4:If aj is absolutely convergent, that is, 1a11 < oo, thenthe seriesa1 is convergent to a fInite sum 1=1

Theorem 5: If a1 4 0, then a convergent series, with sums, say, andfor eachn,

Exercise

6. Let 4) be a monotone function defined on [a,bl Show that and çb(b—)

exist

1.4 FUNCI1ONSAND MAPPINGS

Functions considered will be real-valued (or, briefly, in Chapter 10, valued) functions on some space X In many cases the space X will be R if thefunction f is defined on X and takes its values in Y we will frequently use thenotation f: X -÷ Y. If g: X Y and f: Y -÷ Z, then the composite functionfog: X-÷Zis defined by =f(g(x))

complex-The domain of f is the set [x: 1(x) is definedj complex-The range of f is the set[y: y =f(x) for some xJ If f: X Yand A Y we writef1 (A) Ix: x EX,

fix)E A], and ifB X we write flB)= [y:y = fix),x E B] Thefunction f is

a one4o-one mappingofX onto Y if the domain off is X, the range off is Y,and /1:x1) =f(x2)only if x1 x2 hi this case f1 is a well-defined function on

Y Theidentity mapping is denoted by 1 so I x =x.If afunctionis a one-to-one

mapping, then on the domain of f,f1 of= I and on the range off,fof' = 1.

The function f extends the function g or is an extension of g if the domain of

f contains that of g, and on the domain of g, f =g.Frequently, in applications,

Definition 3: n 1,2, . .,andfbe functions on the space I.

Then 1,, funiformly ifgiven >0, there exists n0E N such that

sup x El] <eforn>n0.

Elementary results concerning continuity and differentiation will be used asrequired, as will the definitions and more familiar properties of standard func-tions A standard result on continuous functions which we will assume known isthe following

Theorem6: Let be a sequence of continuous functions, X -÷ Aand let

1,, uniformly; then f is a continuous function

Statements about sets can be turned into ones about functions using thefollowing notation

Definition 4: Let the set A be contained in the space X; then the characteristicfunction of A, written is the function on X defmed by: = 1 for

Astep function on R is one of the form a1 br,, where =1, ndenotedisjoint intervals An example of such a function which will be used Is sgn xwhich is defined as: sgn x = 1 for x> 0, sgn 0 = 0,sgn x =—1 for x <0

1.5 CARDINAL NUMBERS AND CARDINALITY

Two sets A and B are said to be equipotent, and we write A B,if there exl3ts

a impping with domain A and range B A standard result of set theory(cf [11], p 99) is that with every set A we may associatea well-deflned object,Card A, such that A "-'Bif, and only if, Card A =CardB We say that Card A>Card B if for some A' C A we have B "-'A' but there is no set B' C B such that

A ".'B' We assume the result: for any set A, Card A <Card If Card A =a,

we write Card P (A) =2a.

IfA is fInite, we have Card A =n, the number of elements in A If A N

we write Card A = and describe A as an infInite countable set If A Rwewrite Card A =c.It is easy to show that if Jig any interval, open, closed, or half-open, and if I contains more than one point, then Card I =c.Another result isthat if [Aj: I ENJ is a collection of countable sets, then Card L) A, = Also

1=1

Sec 1 6J Further Properties of Open Sets 23

Card 0 = In Chapter 2, Exercise 45, we need the following result: Card[f: f: A -+Rj =2C; foran explicit proof see

, p.50

1.6 FURThER PROPERTIES OF OPEN SETS

Theorem 7 (LindelOf's Theorem): If = [Ia: a E AJ is a collection of openintervals, then there exists a subcollection, say [Ii: i = 1, 2, . . ], at mostcountable in number, such that UI,= U

Proof Eachx E 'a is contained in an open interval J, with rational end-points,such that J1 Ills for each a; since the rationals are countable the collection[J,] is at most countable Also, it is clear that U 'a= U J,. For each i choose

anintervall1 of such thatl1 Jj Then U 4 = U J, U so we get the

identityand the result 0

If the subcollection obtained is finite, we make the obvious changes ofnotation

Theorem 8 (LindelOf's Theorem in Rn): If = [Ga: a E is a collection ofopen sets in R", then there exists a subcollection of these, say [G,: i = 1,2, .1'

at most countable in number, such that UG1 = U Ga.

Since [t: It—xI<rJ C Ga for some r> 0, there exists an open 'cube'with the sides of length such that x E Ta C Ga, and a 'rectangle' J,with rational coordinates for its vertices and containing x may be chosen within

Ta The proof then proceeds as in R (We have written lx —yI for the usualdistance between points x ,y of R".) 0

1'heorezn 9: Each non-empty open set G in R isthe union of disjoint open vals, at most countable in number

inter-Proof: Following Definition 1, p 16, write a "-'b if the closed interval [a,b], or[b,aJ if b <a, liesin G This is an equivalence relation, in particular a "-' a since [aJ is itself a closed interval G is therefore the union of disjoint equivalenceclasses Let C(a) be the equivalence class containing a Then C(a) is clearly aninterval Also C(a) is open, for if k E then (k — e,k + e) ç G for suffi-ciently small e But then (k e, k + e) c so G is the union of disjointintervals These are at most countable in number by Theorem 7.0

1.7 CANTOR-LIKE SETS

We now describe theCantor-like sets These, and the functions defined on them,

24 Preliminaries [Oi I

are particularly useful for the construction of counter-examples A special case—

the Cantor ternary set or Cantor set —is sufficient for many purposes and will

be described separately

The construction is inductive From [0,11 remove an open Interval with

centre at 1/2 and of length < 1 This leaves two 'residual intervals', J11, J1,2,each of length < 1/2 Suppose that the nth step has been completed, leavingclosed intervals .,J,,2n,each of length < 1/2" We carry out the (n + l)ststep by removing from each anopen interval ,k withthe same centre as

2"

k and of length < 1/2" Let 1',, = U andlet P= fl P,1. Any set P

formedin this way is a Cantor-like set

In particular P contains the end-points of each J,, Since [0,11 — P =

lie in (x — €, x+ e) But these end-points belong to P, so x is a limit point of P

A particular case which will be useful is when 1(J11) =l(J1,2) = < 1/2,l(J2,1) = . = = etc., where 1(1)denotes the length of the interval I

So at each stage residual interval is divided in the same proportions as the originalinterval [0,11 Denote the resulting Cantor-like set, in this case, by Slightlymore generally, let 1(J11) =l(J1,2) = 1/2), l(J2,1) = . = =etc., so that at each stage the residual intervals are equal but the proportions areallowed to change from stage to stage Denote the resulting set in this case by Pt,where = fbi, . ).Note that for each n Use is made of Ptand in the next chapter

We may vary the construction by choosing the removed open intervalscentre', with centres a fIxed combination y, 1 — of the of the

where 0 <y < 1 For a general construction of such perfect sets, includingthose given, see [7], Chapter 1.

The Cantor Set P

From the interval [0,1] first remove (1/3, 2/3), then (1/9, 2/9) and (7/9,8/9),etc., removing at each stage the open intervals constituting the 'middle thirds'

of the closed intervals left at the previous stage This gives a special case of theprevious constructions, with the residual closed intervals at the nth stage, J,,1, ,each of length 1/3", the open intervals Ifl, also being of length 1/3" if

Sec 1.7] Sets 25

below or can be seen directly as follows It consists of those points x which can

be given an expansion to the base 3 in the form x =O.x1x2 with =0 or 2.Suppose that P is countable and let ., be an enumeration of P If

=0,let = 2;if = 2, let x =0.x1x2 . differsfromeach butx EP So no enumeration exists

Example 4: Every non-empty perfect set E ç R is uncountable

Solution: Suppose false, so E may be enumerated as a sequence {xfl }.Form thesequence {yn} in E inductively as follows Let Yi =X1 , =x2, and choose

0 <1x1 —x2 I SmceEisperfectwemaychoosey3 EE,y3 EN(x2,e1),

in the notation of p 17, # x3 ,and with a neighbourhood N(y3 ,e2) C N(x2 ,e1)

We may suppose that 0 <e2 <e1/2 and that x1, x2, x3 are not in N(y3,e2).Now choose y4 E E, y4 E N(y3 ,e2), with a neighbourhood N(y4 ,e3) C N(y3 ,e2),

such that 0 <e3 <€212 and that x1, ., x4 N(y4,e3), etc., by induction.Then (y,11 is a Cauchy sequence in E with limit Yo and Yo E E as any perfectset is closed But for each n, N(yfl, efl_1)containsYo but does not contain

So Yo forany n, and so no such enumeration of E exists

Clearly the result and its proof apply in any complete metric space

The Lebesgue function

For each n, let be the monotone increasing function on [0,11 which is linearand increasing by 1/2" on each and constant on the removed intervals'n,k, where the notations 'n,k' refer to the construction of the Cantor set

P So 0, = 1. It is easy to see, from a diagram, that for n > m,

— Lm(X)1< WriteL(x) = Urn Then the Lebesgue function

26 Preliminaries I

We may similarly construct the Lebesgue function corresponding to theCantor-like set Pt As before the function, again denoted by L is continuous andmonotone increasing The expression for L(x) corresponding to (1.2) is now

more complicated

Example 5: Consider the special case of Cantor-like sets such that for some fixed

a, 0 <a < 1, we have in an obvious notation, = ct/3", for k = 1,

and for each n Denote the set obtained in this way by F(a), theCantor set P being obtained for a = 1.Show that there is a continuous increasingfunction F on [0,1] such that =P.

Solution: Let F,, be the monotone increasing piecewise-linear function, F,,:[0,1J [0,1], mapping the end-points of the onto those of /,, for

in fact — Fm I k1m,k) = 1/3m So for each x, LFm(X)) converges and

urn Fm(x)defmes a function Fon [0,1] Since (x)—F(x)I= lim —

Fm(X)1 < 1/3m, tFmI converges uniformly to F So F is continuous and isclearly monotone increasing We have F([O,1J) = [0,1] and = foreach n and k, so F(F@)) =P. We need only show that F is one-toone Suppose(x,y) C [0,1] if either x or y lies in a removed interval It is easy to see thatF(y) > F(x) So suppose x and y lie in than as is nowhere dense, there

is an interval 'k c (x,y) for some n,k But then F(y) —F(x)> 1/3",so F isstrictly increasing

Exerci9es

7 Find the length of the intervals of Example 5

8 Using the notation of the construction of the Lebesgue function, p 25, showthat the estimate of ILm —LI may be improved to ILm —LI 1/3.2m

CHAPTER 2

Measure on the Real Line

We consider a class of sets (measurable sets) on the real line and the functions(measurable functions) arising from them It is for this large class of functionsthat we will construct a theory of integration in the next chapter On this class

of sets, which includes the intervals, we show how to define Lebesgue measurewhich is a generalization of the idea of length, is suitably additive and is invariantunder translations of the set Apart from integration theory the methods are ofindependent interest as tools for studying sets on the real line Indeed for setswhich are 'scanty' we further refine the idea of measure in the last section §2.6and construct Hausdorff measures particularly appropriate for the Cantor-likesets constructed in the last chapter Sections 2.5 and 2.6 will not be used in theintegration theory of the next chapter

2.1 LEBESGIJEOUTERMEASURE

All the sets considered in this chapter are contained in R, the real line, unlessstated otherwise We wifi be concerned particularly with intervals I of the form

I = [a,b), wherea and b are finite, and unless otherwise specified, intervals may

be supposed to be of this type When a =b, I is the empty set 0 We will write1(f) for the length of I, namely b —a

Definition 1: The Lebesgue outer measure, or more briefly the outer measure, of

a set is given by m(A) =inf where the infimum is taken over all finite

or countable collections of intervals [I,,J such that A c U

For notational convenience we need only deal with countable coverings ofA; the finite case is included since we may take =0except for a finite number

28 Measure onthe Real Line [th.2

Proof: (i), (ii) and (iii) are obvious Since x I,, [x, x + (1/n)) for each n,and = 1/n,(iv) follows 0

Example 1: Show that for any set A, m*(A) = m*(A + x) where A + x =[y+ x: y , thatis: outer measure is translation invariant

Solution: For each e > 0 there exists a collection [Ia] such that A ç andm*(A) 1(1,,) — e.But clearly A + x c + x) So, for each e, m(A +x)

Theorem 2: The outer measure of an interval equals its length

Proof: Case 1 Suppose that I is a closed interval, [a,bJ, say Then, for each

e >0, we have from Theorem 1 and Definition 1 that

Theorem, p 18, a finite subcollection of the 1,, say ., where 1k =

(Ck,dk), covers I Then, as we may suppose that no 1k iscontained in any other,wehave,supposingthatc1 <c2 < <CN,

Case 2 Suppose that I =(a,!,),and a > If a = b, Theorem 1(u) gives theresult.Ifa<b,suppose thatO<e<b —aandwritel'= [a+e,bl.Then

Sec 2.1] LebesgueOuter Measure 29Suppose thatI = al, the other cases being similar Forany M> 0, thereexists k such that the finite interval 'M,where'M= [k,k + M), is contained in I.Som*(1)>Mandhencem*(1)=oo=l(1) D

The next theorem asserts that m *hasthe property of countable subadditivity.Theorem 3: For any sequence of sets ,m*(U E,) m*(E,)

Proof: For each i, and for any e> 0, there exists a sequence of intervals

But isarbitrary and the result follows D

Example 2: Show that, for any set A and any €> 0,there is an open set 0 tainingA and such that m*(O)<m*(A) + €

con-Solution:Choose a sequence such that A ç Li I,,and 1(1,,) —

Exampk3: Suppose that in thedefInition of outer measure, m*(E) =inf 1(1,,)

for sets E R, we stipulate (i) open, (ii) I,, = [an,ba), (iii) I,, =(an, ba],

(iv) I,, closed, or (v) mixtures are allowed, for different n, of the various types ofinterval Show that the same m* is obtained

Solution: In case (II) we obtainthe m* of Definition 1, p 27 Write the ponding m* as in case (i), in case (iii), in case (iv), m7,1 in case (v)

cones-We show that each equals Consider the proof In the other cases beingsimilar Fromthe definition, < To prove the converse: for each

€> 0 and each interval I,, let be an open interval containing I,, with =

(1 + Suppose that the sequence is such that E ç U and

n1

L 1(1,,)—€ Then + )(1 + But E ç U a union of

30 Measure on the Real Line [(1.2open intervals, so (1 + e) +e(l + e), for any >0,so <

asrequired

Exercises

1 Show that if m*(A) 0, then m*(A UB)= m*(B) for anysetB

2 Show that every countable sethas measurezero.

3 Let be a finite set ofintervalscovering the rationals in [0,11 Show that

1.

4 Show that the intervals of Example 3 may be restricted so as to haveendpoints in some set dense in R, for example the rationals, and again ineach case the function m* obtained is unaltered

2.2 MEASURABLE SETS

Definition 2: The set E is Lebesgue measurabLe or, more briefly, measurable iffor each set A we have

As m* issubadditive, to prove Eismeasurable we need only show, for each A,that

Solution: By Theorem 1(111), p 27,(2.6)is satisfied for each A

Definition 3: A class of subsets of an arbitrary space X is said to be a a-algebra(sigma algebra) or, by some authors, a a-field, if X belongs to the class andthe

class is closed under the formation of countable unions and of complements.Definition 4: If in Definition 3 we consider only finite unions we obtain analgebra (or a field)

We will denote by the class of Lebesgue measurable sets

Theorem 4: The class Q5)! isa a-algebra

Proof: From Definition 2, R E and the symmetry in Defmition 2 between

EandCE implies that if E E then CE E Soit remains to be shown that if

{E1} is a sequence of sets in then E =U E,E

1=1 Let A bean arbitrary set By (2.5) (with E replaced by E1) we have

m*(A)=m*(A flCE1),

andby (2.5) again (with Ereplacedby E2 and A by A fl CE1) we have

m*(A)=m*(A nE2 flCE1)+m*(A

Sec 2.2] Measurable Sets 31

Continuingin this way we obtain, for n 2,

usingTheorem 3 twice, and using the fact that for any n, U(E1 =

U E1 Hence we have equality throughout in (2.7) and we have shown that

thatis, m is countably additive on disjoint sets of

hvof: Take A = U E1 in (2.7) which we have seen to be an equality, and note

1=1thatthe expression simplifies since the sets E1 are disjoint 0

Note that since the sets E1 in (2.8) may be replaced by 0froma certain stageonward, the saim result for finite unions follows as a special case

If E is a measurable set we will write m(E) in place of m*(E) Then the set

that m is a countably additive set function, and m(E) is called the Lebesgue

measureof E

32 MeasuxeontheRealLine Ith.2

Theorem 6: Every interval is measurable

Proof: We may suppose the interval to be of the form [a,oo), as Theorem 4 thengives the result for the other types of interval For any set A wewish to showthat

m(A) m(A fl (—oo,a)) + m(Afl [a, oo)) (2.9)

Write =A fl a)and A2 = A C) Ia, co) Then for any e>0 there existintervals I,,suchthat Ac U I,, andm(A)> 1(1,,) — e Write i, =

anamely the class of all subsets of X So taking the intersection of the

au-algebra, necessarily the smallest, containing 0This theorem holds with 'a-algebra' replaced by 'algebra', the class obtainedbeing the generated algebra The proof is the same

Definition 5:We denote by the a-algebra generated by the class of intervals ofthe form Ia,b); its members are called the Borelsetsof R

Theorem 8: (i) ç that is every Borel set is measurable

(ii) 6/3 is the a-algebra generated by each of the following classes:the open intervals, the open sets, the -sets, the Fe-sets

Proof: (i) follows immediately from Theorem 4, p 30, and Theorem 6, p 32.(ii) Iet B be the a-algebra generated by the open Intervals Every openinterval, since it is the union of a sequence of intervals of the form Ia,b), is aBorel set So ç But every interval [a,b),is the intersection of a sequence

of open intervals and so 6/3 c So = Since every open set is theunion of a sequence of open intervals the second result follows Since G6-sets

and F0-sets are from open sets using only countable intersections andcomplements the result in these cases follows similarly 0

Definition 6: For any sequence of sets tE, }

It is easily seen from the definition that lim inf E, ç lim sup E, If they areequal, this set is denoted by lim E, It also follows from the definition thatlhn sup E1 is the set of points belonging to infinitely many of the sets E, and thattim inf is the set of points belonging to all but finitely many of the sets It

is also immediate that if E1 ç E2 ç , thenurn E, U E, and that if E1

1=1 E2 , then lim E, = fl E,, which is analogous to the result that monotone

1=1 sequences of numbers have limits

Theorem 9: Let {E, } bea sequence of measurable sets Then

(i) if E1 ç , we have m(lirn E1) =Jimm(E,),

(ii) if E1 E2 , and m(E,) <oofor each I, then we have m(lim E,) =

(ii) We haveE1 —E1 çE1 —E2 çE1 —E3 ,soby(i)

mQiin(E1 —E,))=lim(m(E1 —E,))

But urn (E1 —E1)= U (E1 —E,)=E1 — E, = E1 — JimE, So taking the

34 Measure on the Real Line (Cli 2measures of both sides the result follows from (2.10) since m(E1) <oo 0

Example 7: (i) Show that every non-empty open set has positive measure

(ii) The rationals Q are enumerated as q1, q2, , and the set G

is defined by

Prove that, for any closed set F, m(G F)> 0

Solution (i) follows immediately from Theorem 9, p 23, and Theorem 2, p 28.(ii) if m(G —F)> 0 there is nothing to prove If m(G —F)=0,then since

G — F is open we must, by (1), have G F But G contains Q whose closure

is R, soF= Rand m(F)=oo But m(G)<2 1/n2 <°°. So m(F— G)

'1=1

andthe result follows

Example 8: Show that there exist uncountable sets of zero measure

Solution: We show that the Cantor set P, p 24,is measurable and m(P) 0 Fromthe construction the sets P,, are measurable for each n, soP = fl F,, is

8 For k> 0 and A ç R let kA = [x: k1x E Al Show that (i) m(kA)

km *(A), (ii) A is measurable iff kA is measurable

9 For A ç R let —A = [x: —x E A] Show that (i)m*(A) =m*(_A),(ii)A

is measurable iff —A is measurable

10 Let E ç M where Mis measurable and m(M) < Showthat E is measurableiffm(M) = m*(E)+m*(M—E).

14 Show that each Cantor-like set has measure zero.

15 Show that the Cantor-like is measurable with measure 1 —a

16 Let G be the set of numbers which can be represented in the form

—+—+ +—+

5

where c,, 0 or 4 for each n Show that m(G) = 0.

17 Show that the set of numbers in [0,11 which possess decimal expansionsnot containing the digit 5 has measure zero

18 Let k be a positive integer and {n1} a finite sequence of positive integers, allless than k Show that the set of numbers in [0,11, in whose expansions tobase k the sequence {n1 } doesnot occur, has measure zero

19 Give an example of a set A ç [0,11 such that m*(A)> 0 and m*(A flT)<

1(1)forall open intervals I ç [0,11

20 Show that [0,11 may be written as the union of a countable number ofnowhere dense perfect sets and a set of measure zero

21 Find an upper bound for the number of sets in the a-algebra generated by

n sets

22 I.et 9 be a u-algebra containing an infinite number of distinct sets Showthat 9 contains an uncountable number of sets

23 Let S be a bounded set Show that every real number is the mid-point of

an open interval such that S flIand S flC!have outer measure

2.3 REGULARITY

Thenext results states that the measurable sets are those which can be mated closely, in terms of m*, by open or closed sets A countablyadditive set function satisfying the conditions (ii) to (iii)* below is said to be aregular measure For the terminology G6, F0, used in the theorem, see Chapter 1,

approxi-p 18

Theorem 10:The following statements regarding the set E are equivalent:(i) E is measurable,

36 Measure on theRealLine [Ch 2(ii) y e>O, 0, anopen set, 0 Esuch

(ifi)

(jj)* Ye>O,

(iil)* F, an Fe-set, F E such that m*(E —F)=0.

Proof: (i) (ii): suppose first that <°°. Asin Example 2 there is an openset 0 E such that m(0) <m(E) + e So m(0 —E)=m(0) m(ff) e.

If oo,write R = U I,,, a union of disjoint finite intervals Then if

n1

E fl I,,, we have m(E,,) < °°so there is an open set 0,, E,, such that

m(0,, —E,,) €/2" Write 0 = U 0,,,an open set Then

for each n, and the result follows

(iii) (1): E = G — (G —E),the set G is measurable and by Example 4, p 30,

G —E is measurable So E is measurable

(i) (ii)*: CE is measurable and so from above there exists an open set 0such that 0 CE and m(0 — CE) e But 0— CE =E— C0,so taking F =COgives the result

(li)* (iii)*: for each n, let F,, be a closed set, F,, c Eand m*(E —F,,)<1/n Then if F = U F,,,F is an Fe-set, F c E, and, for each n, m*(E —F)<m*(E —F,,) < 1/n, and the result follows

(ffl)* (1): since E =FU (E —F),we find E measurable as before DTheorem Ii: If m*(E) <oo thenE is measurable if, and onlyif, V e> 0,joint finite intervals , I,, such that m*(E LJ < e We may stipulatethat the intervals be open, closed or half-open

Proof: Suppose that E is measurable Then by the last theorem ve > 0, 3 an

open set 0 containing E with m(0 —E)< e As m(E) is finite so is m(O) But byTheorem 9, p 23, 0 is the union of disjoint open intervals Ij, I = 1, 2,. So

by Theorem 5, p.3 1, n such that I(I,)< e Write U =U Then E U =

1=1

Sec 2.4] Measurable Functions 37(E—(J)U(U—ff)C_

If we wish the intervalsto be, say, halfopen, we first obtainopen intervals

• , as above and then for each i choose a half-open intervalJ1C suchthat m(Ij —J1)< c/n Then the intervals are disjoint and we have by Example

l,p 16,

so the construction goes through for the intervals

We prove the converse By Example 2, p 29, y >0, 0 open, 0 2 E suchthat

+ e So by (2.11),m(O U)=m(O—U)=m(O)—m(U)<m*(E) m(U) +

Exerci9es

24 Show that if m(E) <00 and there exist intervals - , such that

m(E U < theneach of the intervals is finite

25 Show that the condition m(E) < -isnecessary in Theorem 11, even if thecondition 1(1,) <°°is removed

26 Show that in Theorem 11, n will in general depend on and give an example

tothe real number system with the conventions that

(areal,oraoo),

38 Measure on the Real Line [(1.2

In practice the domain of definition off will usually be either R or R — F

Proof: Let f be measurable Then

Ix:f(xPal = fl [x:f(x)>a— is measurable;

so Let [x:1(x)>a] bemeasurable.Then [x:f(x)<a]

is measurable and (ii) (iii) Jf (iii) holds, then

a measurable set Similarly for a =

Example10: The constant functions are measurable

Solution: Depending on the choice of a, the set [x: 1(x) > a], where f is stant, is the whole real line or the empty set

con-Sec 2.41 Measurable Functions 39Example 11: The characteristic function ofthe set A, is measurable iff A ismeasurable.

Solution: Depending on a, the set [x: (x) > al =A, A or 0, and the resultfollows

Example 12: Continuous functions are measurable

Solution: 1ff is continuous, [x: flx) > a] is open and therefore measurable.Theorem 13: Letc be any real number and let f and g be real.valued measurablefunctions defined on the same measurable set E Then f + c, cf, f + g, f— g and

fg are also measurable

Proof: For each a, [x: f(x) + c > aJ = [x:flx) >a —c], ameasurable set So

f + c is measurable If c 0, cf is measurable as in Example 10 above; otherwise,

if c> 0, [x: cf(x) >al = [x: flx) > c1a1, a measurable set, and similarly for

c <0 So cf is always measurable To show that f+ g is measurable, observe that

x EA = [x:f(x) + g(x) > a] only if f(x) > a —g(x), that is, only if there exists

a rational r1 such thatf(x) > > a —g(x), where fri, 1 1, 2, . } is an

enumera-tion of Q But theng(x) >a —rj and soxEE [x:f(x)>rj] fl [x:g(x)>a —rj].

Hence A B = U ([x: f(x) > rjj [x: g(x) > a — r,]), a measurable set

measurable set The case of f—g is similar 0

Theorem 14: Let {f,,} be a sequence of measurable functions defined on thesame measurable set Then

(i) sup fjismeasurableforeachn,