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MEASURE AND INTEGRATION: LECTURE 1Preliminaries. Weneedtoknowhowtomeasurethesizeorvol-umeofsubsetsofaspaceXbeforewecanintegratefunctionsf:XRorf:X C.

WerefamiliarwithvolumeinRn. WhataboutmoregeneralspacesX? Weneedameasurefunction :{subsetsofX} [0, ].

Fortechnicalreasons,ameasurewillnotbedefinedonallsubsetsofX,butinsteadacertaincollectionofsubsetsofXcalleda-algebra,acollectionofsubsetsofX(i.e.,acollectionM P(X)thatisasubsetofthepowersetofX)satisfyingthefollowing:

.(1) X Mc(2) IfA M,thenA .X\ A M

(3) IfAi M(i=1, 2, . . .),then .i=1 MConstrastwithatopology P(X),whichsatisfies

(1) andX.n(2) IfUi (i=1, . . . , n),theni=1Ui .

(3) IfU ( I)isanarbitrarycollectionin,thenIU .Remarkson-algebras:

(a) By(1),X M,soby(2), M.c(b) Ai =( Ai)c countableintersectionsareinM.i=1 i=1

(c) A, B M A \ B M (sinceA \ B=A Bc).Let(X, X)and(Y, Y)beatopologicalspaces. Thenf:X Y is

continuousiff1(U)X forallU Y. Inverseimagesofopensetsareopen.

Let(X, M)beameasurespace(i.e.,Misa-algebraforthespaceX). Then f: X Y is measurable if f1(U) M for all U Y.Inverseimagesofopensetsaremeasurable.Basicpropertiesofmeasurable functions.Proposition0.1. LetX , Y , Z be topologicalspacessuch that

X f Y g Z. (1) Iff andg arecontinuous, theng f iscontinuous.

Proof. (g f)1(U)=f1(g1(U))=f1(open)=open. Date:September4,2003.

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2 MEASUREAND INTEGRATION: LECTURE 1(2) Iff ismeasurableandgiscontinuous,theng f ismeasurable.

Proof. (g f)1(U)=f1(g1(U))=f1(open)=open. Theorem0.2. Let u :X R, v:X R, and :R R Y. Set h(x) = (u(x),v(x)): X Y. If u and v are measurable and iscontinuous, thenh :X Y ismeasurable.Proof. Define f: X = R2 by f(x) = u(x)v(x). Then R Rh = f. Wejust need to show (NTS) that f is measurable. LetR R2 bearectangleoftheformI1 I2 whereeachIi R(i=1,2)isanopeninterval. Thenf1(R)=u1(I1) v1(I2). Letxf1(R)sothatf(x) R. Thenu(x) I1 andv(x) I2. Sinceu ismeasurable,u1(I1) M,andsince v ismeasurable, v1(I2) M. SinceM isa-algebra,

u1

(I1) v

1

(I2) M.

Thus

f1

(R) Mfor

any

rectangle

R. Finally, any open set U = (rectangle around points withi=1Rirationalcoordinates). Sof1(U)=f1( Ri)= f1(Ri). Eachi=1 i=1termintheunionisinM,sosincecountableunionsofelementsinMareinM . ,f1(U) MExamples.

(a) Letf:X Cwithf =u+iv andu,v realmeasurable func-tions. Thenf iscomplexmeasurable.

(b) If f = u+iv is complex measurable on X, then u,v, and |f|are real measurable. Take to be z Rez, z Imz, and

z,respectively.z | |(c) If f,g are real measurable, then so are f +g and f g. (Alsoholdsforcomplexmeasurablefunctions.)

(d) IfE X ismeasurable(i.e.,E M),thenthecharacteristicfunctionofE,

E(x)= 1 ifxE;0 otherwise.

Proposition0.3. LetF beanycollectionofsubsetsofX. Thenthereexists a smallest -algebra M such that F M. We call M the-algebrageneratedbyF.Proof. Let=thesetofall -algebrascontainingF. ThepowersetP(X)=thesetofallsubsetsof X isa-algebra, so isnotempty.DefineM = . SinceF M forallM ,wehaveF M.MMIfMisa-algebracontainingF,thenM Mbydefinition. Claim:

is a -algebra. If AM, take M. M is a -algebra andM. Thus,Ac M,andsoAc sinceM . IfAi MA M M M

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3MEASUREAND INTEGRATION: LECTURE 1foreach i= 1, 2, . . .,then A M, andsoiAi . It followsthat M

.iAi M

BorelSets. Bythepreviousproposition, ifX isatopologicalspace,thenthereexistsasmallest-algebraBcontainingtheopensets. Ele-mentsofBarecalledBorelsets.

If f: (X, B) (Y, ) and f1(U) B for all U , then f iscalledBorel measurable. Inparticular,continuous functionsareBorelmeasurable.

Terminology: F (F-sigma)=countableunionofclosedsets. G (G-delta)=countableintersectionofopensets.