LECTURE NOTES IN MEASURE THEORY - Chalmers

1 1 LECTURE NOTESIN MEASURE THEORYC hrister BorellMatematikChalmers och G teborgs universitet412 96 G teborg(Version: January 12)2 PREFACET hese are LECTURE NOTES on integration THEORY for a eight-week course at theChalmers University of Technology and the G teborg University. The partsde ning the course essentially lead to the same results as the rst threechapters in the Folland book[F];which is used as a text book on the proofs in the LECTURE NOTES sometimes di er from those given in[F]:Hereis a brief description of the di erences to simplify for the Chapter 1 we introduce so called -systems and -additive classes,which are substitutes for monotone classes of sets[F]. Besides we prefer toemphasize metric outer measures instead of so called premeasures. Through-out the course, a variety of important measures are obtained as image mea-sures of the linear MEASURE on the real line.

2 In Section positive measuresinRinduced by increasing right continuous mappings are constructed in 2 deals with integration and is very similar to[F]and mostother 3 starts with some standard facts about metric spaces and relatesthe concepts to MEASURE THEORY . For example Ulam s Theorem is existence of product measures is based on properties of -systems and -additive 4 deals with di erent modes of convergence and is mostly closeto[F]:Here we include a section about orthogonality since many studentshave seen parts of this THEORY Lebesgue Decomposition Theorem and Radon-Nikodym Theoremin Chapter 5 are proved using the von Neumann illustrate the power of abstract integration these NOTES contain severalsections, which do not belong to the course but may help the student to abetter understanding of MEASURE THEORY . The corresponding parts are setbetween the symbols###and""" I would like to express my deep gratitude to the students inmy classes for suggesting a variety of improvements and a special thankto Jonatan Vasilis who has provided numerous comments and corrections inmy original teborg 2006 Christer Borell4 CONTENT1 -Algebras and MEASURE Determining Lebesgue Carath odory s Existence of Linear Measure2 Integration of Functions with Values in[0;1] Integration of Functions with Arbitrary Comparison of Riemann and Lebesgue Integrals3 Further Construction Methods of Metric Linear Functionals and q-Adic Expansions of Numbers in the Unit Product Change of Variables in Volume Independence in Probability4 Modes of Convergence in MEASURE , inL1( ).

3 And inL2( ) The Haar Basis and Wiener Measure5 Decomposition of Complex The Lebesgue Decomposition and the Radon-Nikodym The Wiener Maximal Theorem and Lebesgue Di erentiation Absolutely Continuous Functions and Functions of Bounded Conditional Expectation6 Complex Complex The Fourier Fourier Non-Di erentiability of Brownian PathsReferences6 CHAPTER 1 MEASURESI ntroductionThe Riemann integral, dealt with in calculus courses, is well suited for com-putations but less suited for dealing with limit processes. In this course wewill introduce the so called Lebesgue integral, which keeps the advantages ofthe Riemann integral and eliminates its drawbacks. At the same time we willdevelop a general MEASURE THEORY which serves as the basis of contemporaryanalysis and this introductory chapter we set forth some basic concepts of measuretheory, which will open for abstract Lebesgue -Algebras and MeasuresThroughout this courseN=f0;1;2;:::g(the set of natural numbers)Z=f:::; 2; 1;0;1;;2;:::g(the set of integers)Q=the set of rational numbersR=the set of real numbersC=the set of complex R; A+is the set of all strictly positive elements inA:Iffis a function from a setAinto a setB;this means that to everyx2 Athere corresponds a pointf(x)2 Band we writef:A!

4 B:A function isoften called a map or a mapping. The functionfis injective if(x6=y))(f(x)6=f(y))7and surjective if to eachy2B;there exists anx2 Asuch thatf(x) =y:An injective and surjective function is said to be setAis nite if eitherAis empty or there exist ann2N+and abijectionf:f1;:::;ng !A:The empty set is denoted by :A setAis saidto be denumerable if there exists a bijectionf:N+!A:A subset of adenumerable set is said to be at most a set. For anyA X;the indicator function AofArelativetoXis de ned by the equation A(x) = 1ifx2A0ifx2Ac:The indicator function Ais sometimes written1A:We have the followingrelations: Ac= 1 A A\B= min( A; B) = A Band A[B= max( A; B) = A+ B A B:De nition a ) A collectionAof subsets ofXis said to be an algebra inXifAhasthe following properties:(i)X2 A:(ii)A2 A)Ac2 A;whereAcis the complement ofArelative toX:(iii) IfA;B2 AthenA[B2 A:(b) A collectionMof subsets ofXis said to be a -algebra inXifMis an algebra with the following property:IfAn2 Mfor alln2N+, then[1n=1An2 M:8 IfMis a -algebra inX;(X;M)is called a measurable space and themembers ofMare called measurable sets.]]]

5 The so called power setP(X),that is the collection of all subsets ofX, is a -algebra inX:It is simple toprove that the intersection of any family of -algebras inXis a -algebra. Itfollows that ifEis any subset ofP(X);there is a unique smallest -algebra (E)containingE;namely the intersection of all -algebras containingE:The -algebra (E)is called the -algebra generated byE:The -algebragenerated by all open intervals inRis denoted byR. It is readily seen thatthe -algebraRcontains every subinterval ofR. Before we proceed, recallthat a subsetEofRis open if to eachx2 Ethere exists an open subintervalofRcontained inEand containingx;the complement of an open set is saidto be closed. We claim thatRcontains every open subsetUofR:To seethis supposex2 Uand letx2]a;b[ U;where 1< a < b <1:Nowpickr;s2 Qsuch thata < r < x < s < b:Thenx2]r;s[ Uand it followsthatUis the union of all bounded open intervals with rational boundarypoints contained inU:Since this family of intervals is at most denumberablewe conclude thatU2 R:In addition, any closed set belongs toRsince itscomplements is open.

6 It is by no means simple to grasp the de nition ofRatthis stage but the reader will successively see that the -algebraRhas verynice properties. At the very end of Section , using the so called Axiom ofChoice, we will exemplify a subset of the real line which does not belong toR. In fact, an example of this type can be constructed without the Axiomof Choice (see Dudley s book[D]).In MEASURE THEORY , inevitably one encounters1:For example the realline has in nite length. Below[0;1] = [0;1[[f1g:The inequalitiesx yandx < yhave their usual meanings ifx;y2[0;1[. Furthermore,x 1ifx2[0;1]andx <1ifx2[0;1[:We de nex+1=1+x=1ifx;y2[0;1];andx 1=1 x= 0ifx= 01if0< x 1:Sums and multiplications of real numbers are de ned in the usual X; n2N+, andAk\An= ifk6=n, the sequence(An)n2N+iscalled a disjoint denumerable collection. If(X;M)is a measurable space, thecollection is called a denumerable measurable partition ofAifA=[1n=1 AnandAn2 Mfor everyn2N+:Some authors call a denumerable collectionof sets a countable collection of nition (a) LetAbe an algebra of subsets ofX:A function :A!]]]]]]]]

7 [0;1]is called a content if(i) ( ) = 0(ii) (A[B) = (A) + (B)ifA;B2 AandA\B= :(b) If(X;M)is a measurable space a content de ned on the -algebraMis called a positive MEASURE if it has the following property:For any disjoint denumerable collection(An)n2N+of members ofM ([1n=1An) = 1n=1 (An):If(X;M)is a measurable space and the function :M ![0;1]is apositive MEASURE ,(X;M; )is called a positive MEASURE space. The quantity (A)is called the - MEASURE ofAor simply the MEASURE ofAif there isno ambiguity. Here(X;M; )is called a probability space if (X) = 1;a nite positive MEASURE space if (X)<1;and a - nite positive measurespace ifXis a denumerable union of measurable sets with nite MEASURE is called a probability MEASURE , nite MEASURE , and - nitemeasure, if(X;M; )is a probability space, a nite positive MEASURE space,and a - nite positive MEASURE space, respectively.]]

8 A probability space isoften denoted by( ;F;P):A memberAofFis called an soon as we have a positive MEASURE space(X;M; ), it turns out tobe a fairly simple task to de ne a so called -integralZXf(x)d (x)as will be seen in Chapter class of all nite unions of subintervals ofRis an algebra which isdenoted byR0:IfA2R0we denote byl(A)the Riemann integralZ1 1 A(x)dxand it follows from courses in calculus that the functionl:R0![0;1]is acontent. The algebraR0is called the Riemann algebra andlthe Riemanncontent. IfIis a subinterval ofR,l(I)is called the length ofI:Below wefollow the convention that the empty set is an P(X),cX(A)equals the number of elements inA, whenAis a nite set, andcX(A) =1otherwise. Clearly,cXis a positive MEASURE . ThemeasurecXis called the counting MEASURE onX:Givena2X;the probability MEASURE ade ned by the equation a(A) = A(a);ifA2 P(X);is called the Dirac MEASURE at the pointa:Sometimeswe write a= X;ato emphasize the setX:If and are positive measures de ned on the same -algebraM, thesum + is a positive MEASURE onM:More generally, + is a positivemeasure for all real ; 0:Furthermore, ifE2 M;the function (A) = (A\E); A2 M;is a positive MEASURE .

9 Below this MEASURE will bedenoted by Eand we say that Eis concentrated onE:IfE2M;the classME=fA2M;A Egis a -algebra of subsets ofEand the function (A) = (A),A2 ME;is a positive MEASURE . Below this MEASURE will bedenoted by jEand is called the restriction of toME:LetI1;:::;Inbe subintervals of the real line. The setI1 ::: In=f(x1;:::;xn)2Rn;xk2Ik; k= 1;:::;ngis called ann-cell inRn; its volume vol(I1 ::: In)is, by de nition, equaltovol(I1 ::: In) = nk=1l(Ik):IfI1;:::;Inare open subintervals of the real line, then-cellI1 ::: Iniscalled an openn-cell. The -algebra generated by all openn-cells inRnisdenoted byRn:In particular,R1=R. A basic theorem in MEASURE theorystates that there exists a unique positive measurevnde ned onRnsuch thatthe MEASURE of anyn-cell is equal to its volume. The measurevnis called thevolume MEASURE onRnor the volume MEASURE onRn:Clearly,vnis - measurev2is called the area MEASURE onR2andv1the linear measureonR:11 Theorem volume MEASURE will be proved in Section in the special casen= 1.

10 Thegeneral case then follows from the existence of product measures in An alternative proof of Theorem will be given in Section Assoon as the existence of volume MEASURE is established a variety of interestingmeasures can be we prove some results of general interest for positive an algebra of subsets ofXand a contentde ned onA. Then,(a) is nitely additive, that is (A1[:::[An) = (A1) +:::+ (An)ifA1;:::;Anare pairwise disjoint members ofA:(b)ifA;B2A; (A) = (AnB) + (A\B):Moreover, if (A\B)<1;then (A[B) = (A) + (B) (A\B)(c)A Bimplies (A) (B)ifA;B2A:(d) nitely sub-additive, that is (A1[:::[An) (A1) +:::+ (An)ifA1;:::;Anare members ofA:If(X;M; )is a positive MEASURE space12(e) (An)! (A)ifA=[n2N+An; An2M;andA1 A2 A3 ::: :(f) (An)! (A)ifA=\n2N+An; An2M;A1 A2 A3 :::and (A1)<1:(g) is sub-additive, that is for any denumerable collection(An)n2N+ofmembers ofM, ([1n=1An) 1n=1 (An):PROOF (a) IfA1;:::;Anare pairwise disjoint members ofA; ([nk=1Ak) = (A1[([nk=2Ak))= (A1) + ([nk=2Ak)and, by induction, we conclude that is nitely additive.]]]]]]]]]]]