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1 LECTURENOTES INMEASURETHEORY Christer Borell Matematik Chalmers och Göteborgs universitet 412 96 Göteborg (Version: January 12)2 PREFACE These are lecture notes on integration theory for a eight-week course at the Chalmers University of Technology and the Göteborg University. The parts de…ning the course essentially lead to the same results as the …rst three chaptersintheFollandbook F; whichisusedasatextbookonthecourse. Theproofsinthelecturenotessometimesdi¤erfromthosegiveninF:Here is a brief description of the di¤erences to simplify for the reader. In Chapter 1 we introduce so called -systems and -additive classes, which are substitutes for monotone classes of sets F. Besides we prefer to emphasizemetricoutermeasuresinsteadofsocalledpremeasures. Through- out the course, a variety of important measures are obtained as image mea- sures of the linear measure on the real line. In Section 1.6 positive measures inRinducedbyincreasingrightcontinuousmappingsareconstructedinthis way. Chapter 2 deals with integration and is very similar to F and most other texts. Chapter3startswithsomestandardfactsaboutmetricspacesandrelates the concepts to measure theory. For example Ulam’s Theorem is included. The existence of product measures is based on properties of -systems and -additive classes. Chapter 4 deals with di¤erent modes of convergence and is mostly close to F: Here we include a section about orthogonality since many students have seen parts of this theory before. The Lebesgue Decomposition Theorem and Radon-Nikodym Theorem 2 in Chapter 5 are proved using the von Neumann beautiful L -proof. Toillustratethepowerof abstractintegrationthesenotescontainseveral sections, which do not belong to the course but may help the student to a better understanding of measure theory. The corresponding parts are set between the symbols and """ respectively.3 Finally I would like to express my deep gratitude to the students in my classes for suggesting a variety of improvements and a special thank to Jonatan Vasilis who has provided numerous comments and corrections in my original text. Göteborg 2006 Christer Borell4 CONTENT 1 Measures 1.1-Algebras and Measures 1.2 Measure Determining Classes 1.3 Lebesgue Measure 1.4 Carathéodory’s Theorem 1.5 Existence of Linear Measure 2 Integration 2.1 Integration of Functions with Values in 0;1 2.2 Integration of Functions with Arbitrary Sign 2.3 Comparison of Riemann and Lebesgue Integrals 3 Further Construction Methods of Measures 3.1 Metric Spaces 3.2 Linear Functionals and Measures 3.3 q-Adic Expansions of Numbers in the Unit Interval 3.4 Product Measures 3.5 Change of Variables in Volume Integrals 3.6 Independence in Probability 4 Modes of Convergence 1 2 4.1 Convergence in Measure, inL (); and inL () 4.2 Orthogonality 4.3 The Haar Basis and Wiener Measure 5 Decomposition of Measures 5.1 Complex Measures 5.2 The Lebesgue Decomposition and the Radon-Nikodym Theorem 5.3TheWienerMaximalTheoremandLebesgueDi¤erentiationTheorem5 5.4AbsolutelyContinuousFunctionsandFunctionsofBoundedVariation 5.5 Conditional Expectation 6 Complex Integration 6.1 Complex Integrand 6.2 The Fourier Transform 6.3 Fourier Inversion 6.4 Non-Di¤erentiability of Brownian Paths References6 CHAPTER1 MEASURES Introduction The Riemann integral, dealt with in calculus courses, is well suited for com- putations but less suited for dealing with limit processes. In this course we will introduce the so called Lebesgue integral, which keeps the advantages of theRiemannintegralandeliminatesitsdrawbacks. Atthesametimewewill develop a general measure theory which serves as the basis of contemporary analysis and probability. In this introductory chapter we set forth some basic concepts of measure theory, which will open for abstract Lebesgue integration. 1.1. -Algebras and Measures Throughout this course N =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 numbers R = the set of real numbers C = the set of complex numbers. If AR; A is the set of all strictly positive elements inA: + Iff isafunctionfromasetAintoasetB;thismeansthattoeveryx2A there corresponds a point f(x)2B and we write f : A B: A function is often called a map or a mapping. The functionf is injective if (x6=y)) (f(x)6=f(y))7 and surjective if to each y 2 B; there exists an x 2 A such that f(x) = y: An injective and surjective function is said to be bijective. A set A is …nite if either A is empty or there exist an n 2 N and a + bijection f :f1;:::;ngA: The empty set is denoted by : A set A is said to be denumerable if there exists a bijection f : N A: A subset of a + denumerable set is said to be at most denumerable. Let X be a set. For any AX; the indicator function  of A relative A toX is de…ned by the equation  1 if x2A  (x) = A c 0 if x2A : The indicator function  is sometimes written 1 : We have the following A A relations:  c = 1 A A  = min( ; ) =  A\B A B A B and  = max( ; ) = +   AB A B A B A B: De…nition 1.1.1. LetX be a set. a) A collectionA of subsets of X is said to be an algebra in X ifA has the following properties: (i)X 2A: c c (ii)A2A)A 2A; whereA is the complement of A relative toX: (iii) If A;B2A thenAB2A: (b) A collectionM of subsets of X is said to be a -algebra in X if M is an algebra with the following property: 1 If A 2M for all n2N , then A 2M: n + n n=18 If M is a -algebra in X; (X;M) is called a measurable space and the members of M are called measurable sets. The so called power set P(X), that is the collection of all subsets of X, is a -algebra in X: It is simple to provethattheintersectionofanyfamilyof-algebrasinX isa-algebra. It follows that ifE is any subset ofP(X); there is a unique smallest -algebra (E) containingE; namely the intersection of all -algebras containingE: The-algebra(E) iscalledthe-algebrageneratedbyE: The-algebra generated by all open intervals in R is denoted byR. It is readily seen that the -algebra R contains every subinterval of R. Before we proceed, recall thatasubsetE ofRisopeniftoeachx2E thereexistsanopensubinterval ofR contained inE and containingx; the complement of an open set is said to be closed. We claim thatR contains every open subset U of R: To see this suppose x 2 U and let x 2 a;b  U; where 1 a b 1: Now pickr;s2Q such thatarxsb: Thenx2 r;sU and it follows that U is the union of all bounded open intervals with rational boundary points contained inU: Since this family of intervals is at most denumberable we conclude that U 2R: In addition, any closed set belongs to R since its complementsisopen. Itisbynomeanssimpletograspthede…nitionofRat this stage but the reader will successively see that the-algebraR has very nice properties. At the very end of Section 1.3, using the so called Axiom of Choice, we will exemplify a subset of the real line which does not belong to R. In fact, an example of this type can be constructed without the Axiom of Choice (see Dudley’s book D). In measure theory, inevitably one encounters 1: For example the real line has in…nite length. Below 0;1 = 0;1f1g: The inequalitiesxy and x y have their usual meanings if x;y 2 0;1. Furthermore, x1 if x 2 0;1 and x 1 if x 2 0;1: We de…ne x +1 = 1 +x = 1 if x;y2 0;1; and  0 if x = 0 x1 =1x = 1 if 0x1: Sums and multiplications of real numbers are de…ned in the usual way. IfA X;n2N , andA \A = ifk = 6 n, the sequence (A ) is n + k n n n2N + calledadisjointdenumerablecollection. If(X;M)isameasurablespace,the 1 collection is called a denumerable measurable partition of A if A = A n n=1 and A 2M for every n2N : Some authors call a denumerable collection n + of sets a countable collection of sets.9 De…nition 1.1.2. (a) Let A be an algebra of subsets of X: A function  :A 0;1 is called a content if (i)() = 0 (ii)(AB) =(A)+(B) if A;B2A andA\B =: (b) If (X;M) is a measurable space a content de…ned on the-algebraM is called a positive measure if it has the following property: For any disjoint denumerable collection (A ) of members ofM n n2N + 1 1 ( A ) =  (A ): n n n=1 n=1 If (X;M) is a measurable space and the function  : M 0;1 is a positivemeasure, (X;M;) iscalledapositivemeasurespace. Thequantity (A) is called the -measure of A or simply the measure of A if there is no ambiguity. Here (X;M;) is called a probability space if (X) = 1; a …nite positive measure space if (X) 1; and a -…nite positive measure space ifX is a denumerable union of measurable sets with …nite-measure. The measure  is called a probability measure, …nite measure, and -…nite measure, if (X;M;) is a probability space, a …nite positive measure space, and a -…nite positive measure space, respectively. A probability space is often denoted by ( ;F;P): A memberA ofF is called an event. As soon as we have a positive measure space (X;M;), it turns out to be a fairly simple task to de…ne a so called-integral Z f(x)d(x) X as will be seen in Chapter 2.10 The class of all …nite unions of subintervals of R is an algebra which is denoted byR : If A2R we denote by l(A) the Riemann integral 0 0 Z 1  (x)dx A 1 and it follows from courses in calculus that the function l :R 0;1 is a 0 content. The algebra R is called the Riemann algebra and l the Riemann 0 content. If I is a subinterval of R, l(I) is called the length of I: Below we follow the convention that the empty set is an interval. If A 2 P(X), c (A) equals the number of elements in A, when A is a X …nite set, andc (A) =1 otherwise. Clearly, c is a positive measure. The X X measurec is called the counting measure onX: X Givena2X;theprobabilitymeasure de…nedbytheequation (A) = a a  (a); if A2P(X); is called the Dirac measure at the point a: Sometimes A we write =  to emphasize the setX: a X;a If  and  are positive measures de…ned on the same -algebra M, the sum + isapositivemeasureonM:Moregenerally,  +  isapositive measure for all real ;  0: Furthermore, if E 2M; the function (A) = (A\E); A 2 M; is a positive measure. Below this measure  will be E E denotedby andwesaythat isconcentratedonE: IfE2M; theclass M = fA2M; AEg is a -algebra of subsets of E and the function E (A) =(A), A2M ; is a positive measure. Below this measure  will be E denoted by and is called the restriction of  toM : E jE LetI ;:::;I be subintervals of the real line. The set 1 n n I :::I =f(x ;:::;x )2R ; x 2I ; k = 1;:::;ng 1 n 1 n k k n is called an n-cell in R ; its volume vol(I :::I ) is, by de…nition, equal 1 n to n vol(I :::I ) =  l(I ): 1 n k k=1 IfI ;:::;I are open subintervals of the real line, then-cellI :::I is 1 n 1 n n called an open n-cell. The -algebra generated by all open n-cells in R is denoted byR : In particular, R =R. A basic theorem in measure theory n 1 statesthatthereexistsauniquepositivemeasurev de…nedonR suchthat n n themeasureofanyn-cellisequaltoitsvolume. Themeasurev iscalledthe n n volume measure onR or the volume measure onR : Clearly,v is-…nite. n n 2 The measure v is called the area measure on R and v the linear measure 2 1 on R:11 n Theorem 1.1.1. The volume measure on R exists. Theorem1.1.1willbeprovedinSection1.5inthespecialcasen = 1. The general case then follows from the existence of product measures in Section 3.4. An alternative proof of Theorem 1.1.1 will be given in Section 3.2. As soonastheexistenceofvolumemeasureisestablishedavarietyofinteresting measures can be introduced. Next we prove some results of general interest for positive measures. Theorem 1.1.2. Let A be an algebra of subsets of X and  a content de…ned on A. Then, (a)  is …nitely additive, that is (A :::A ) =(A )+:::+(A ) 1 n 1 n if A ;:::;A are pairwise disjoint members of A: 1 n (b) if A;B2A; (A) =(AnB)+(A\B): Moreover, if (A\B) 1; then (AB) =(A)+(B)(A\B) (c) AB implies (A)(B) if A;B2A: (d) …nitely sub-additive, that is (A :::A )(A )+:::+(A ) 1 n 1 n if A ;:::;A are members of A: 1 n If (X;M;) is a positive measure space12 (e) (A )(A) if A = A ; A 2M; and n n2N n n + A A A ::: : 1 2 3 (f) (A )(A) if A =\ A ; A 2M; n n2N n n + A A A ::: 1 2 3 and (A )1: 1 (g) is sub-additive, that is for any denumerable collection (A ) of n n2N + members of M, 1 1 ( A )  (A ): n n n=1 n=1 PROOF (a) If A ;:::;A are pairwise disjoint members ofA; 1 n n n ( A ) =(A ( A )) k 1 k k=1 k=2 n =(A )+( A ) 1 k k=2 and, by induction, we conclude that is …nitely additive. (b) Recall that c AnB =A\B : NowA = (AnB)(A\B) and we get (A) =(AnB)+(A\B): Moreover, sinceAB = (AnB)B; (AB) =(AnB)+(B) and, if (A\B)1; we have (AB) =(A)+(B)(A\B). (c) Part (b) yields (B) =(BnA)+(A\B) =(BnA)+(A); where the last member does not fall below(A):13 n (d) If (A ) is a sequence of members of A de…ne the so called disjunction i i=1 n n (B ) of the sequence (A ) as k i k=1 i=1 k1 B =A andB =A n A for 2kn: 1 1 k k i i=1 k k ThenB A ; A = B;k = 1;::;n;andB \B =ifi6=j:Hence, k k i i i j i=1 i=1 by Parts (a) and (c), n n n ( A ) =  (B )  (A ): k k k k=1 k=1 k=1 (e) Set B = A and B = A nA for n 2: Then A = B ::::B ; 1 1 n n n1 n 1 n 1 B \B = if i6=j andA = B : Hence i j k k=1 n (A ) =  (B ) n k k=1 and 1 (A) =  (B ): k k=1 Now e) follows, by the de…nition of the sum of an in…nite series. (f) PutC =A nA ; n 1: ThenC C C :::; n 1 n 1 2 3 1 A nA = C 1 n n=1 and(A)(A )(A )1: Thus n 1 (C ) =(A )(A ) n 1 n and Part (e) shows that (A )(A) =(A nA) = lim (C ) =(A ) lim (A ): 1 1 n 1 n n1 n1 This proves (f). (g) The result follows from Parts d) and e). This completes the proof of Theorem Thehypothesis”(A )1”inTheorem1.1.2(f)isnotsuper‡uous. If 1 c isthecountingmeasureonN andA =fn;n+1;:::g;thenc (A ) = N + n N n + + 1 1 1 for all n butA A :::: andc (\ A ) = 0 since\ A =: 1 2 N n n + n=1 n=1 If A;BX; the symmetric di¤erenceAB is de…ned by the equation AB = (AnB)(BnA): def Note that  =j  j: AB A B Moreover, we have c c AB =A B and 1 1 1 ( A )( B ) (A B ): i i i i i=1 i=1 i=1 Example 1.1.1. Let  be a …nite positive measure on R: We claim that to each set E 2 R and " 0; there exists a set A; which is …nite union of intervals (that is, A belongs to the Riemann algebraR ), such that 0 (EA)": To see this let S be the class of all sets E 2 R for which the conclusion c is true. Clearly  2 S and, moreover, R  S: If A 2 R , A 2 R and 0 0 0 c thereforeE 2S ifE2S: NowsupposeE 2S;i2N : Thentoeach" 0 i + i andi there is a setA 2R such that(E A ) 2 ": If we set i 0 i i 1 E = E i i=1 then 1 1 (E( A ))  (E A )": i i i i=1 i=1 Here 1 1 c c 1 E( A ) =fE\(\ A )gfE \( A )g i i i=1 i=1 i i=1 and Theorem 1.1.2 (f) gives that n c c 1 (fE\(\ A )gf(E \( A )g)" i i=1 i i=1 if n is large enough (hint: \ (D F) = (\ D )F): But then i2I i i2I i n n c c n (E A ) =(fE\(\ A )gfE \( A )g)" i i i=1 i=1 i i=115 if n is large enough we conclude that the set E 2 S: Thus S is a -algebra and sinceR S R it follows thatS =R: 0 Exercises 1. Prove that the sets NN =f(i;j); i;j2Ng and Q are denumerable. 2. SupposeA is an algebra of subsets of X and  and  two contents onA such that and(X) =(X)1: Prove that =: 3. Suppose A is an algebra of subsets of X and  a content on A with (X)1: Show that (ABC) =(A)+(B)+(C) (A\B)(A\C)(B\C)+(A\B\C): 4. (a)AcollectionC ofsubsetsofX isanalgebrawiththefollowingproperty: 1 If A 2C; n2N andA \A = if k6=n, then A 2C. n + k n n n=1 Prove thatC is a-algebra. (b)AcollectionCofsubsetsofX isanalgebrawiththefollowingproperty: 1 If E 2C andE E ; n2N ; then E 2C . n n n+1 + n 1 Prove thatC is a-algebra. 1 5. Let (X;M) be a measurable space and ( ) a sequence of positive k k=1 measures onM such that   ::: . Prove that the set function 1 2 3 (A) = lim  (A); A2M k k1 is a positive measure.16 6. Let (X;M;) be a positive measure space. Show that q n n n (\ A )  (A ) k k k=1 k=1 for all A ;:::;A 2M: 1 n 7. Let (X;M;) be a-…nite positive measure space with(X) =1: Show that for anyr2 0;1 there is someA2M withr(A)1: 8. Show that the symmetric di¤erence of sets is associative: A(BC) = (AB)C: 9. (X;M;) is a …nite positive measure space. Prove that j(A)(B)j(AB): 10. LetE = 2N: Prove that c (EA) =1 N if A is a …nite union of intervals. 11. Suppose(X;P(X);)isa…nitepositivemeasurespacesuchthat(fxg) 0 for everyx2X: Set d(A;B) =(AB); A;B2P(X): Prove that d(A;B) = 0 ,A =B; d(A;B) =d(B;A)17 and d(A;B)d(A;C)+d(C;B): 12. Let (X;M;) be a …nite positive measure space. Prove that n n ( A )  (A ) (A \A ) i i 1ijn i j i=1 i=1 for all A ;:::;A 2M and integers n 2: 1 n 13. Let(X;M;)beaprobabilityspaceandsupposethesetsA ;:::;A 2M 1 n P n n satisfy the inequality (A )n1: Show that(\ A ) 0: i i 1 1 1.2. Measure Determining Classes Suppose and areprobabilitymeasuresde…nedonthesame-algebraM, which is generated by a classE: If and agree onE; is it then true that and agree onM? The answer is in general no. To show this, let X =f1;2;3;4g and E =ff1;2g;f1;3gg: 1 Then(E) =P(X): If  = c and X 4 1 1 1 1  =  +  +  +  X;1 X;2 X;3 X;4 6 3 3 6 then = onE and = 6 : Inthissectionwewillproveabasicresultonmeasuredeterminingclasses for -…nite measures. In this context we will introduce so called -systems and -additive classes, which will also be of great value later in connection with the construction of so called product measures in Chapter 3.18 De…nition 1.2.1. A class G of subsets of X is a -system if A\B 2 G for all A;B2G: n The class of all openn-cells inR is a-system. De…nition 1.2.2. A class D of subsets of X is called a -additive class if the following properties hold: (a)X 2D: (b) If A;B2D andAB; thenBnA2D: (c) If (A ) is a disjoint denumerable collection of members of the n n2N + 1 classD; then A 2D: n n=1 Theorem 1.2.1. If a -additive class M is a -system, then M is a - algebra. c PROOF. If A 2 M; then A = XnA 2 M since X 2 M and M is a - additiveclass. Moreover,if (A ) isadenumerablecollectionofmembers n n2N + ofM; c c c A :::A = (A \:::\A ) 2M 1 n 1 n 1 for eachn; sinceM is a-additive class and a-system. Let (B ) be the n n=1 1 disjunctionof (A ) :Then(B ) isadisjointdenumerablecollectionof n n n2N n=1 + 1 1 members ofM and De…nition 1.2.2(c) implies that A = B 2M: n n n=1 n=1 Theorem 1.2.2. Let G be a -system and D a -additive class such that G  D: Then (G)D: PROOF. Let M be the intersection of all -additive classes containing G: TheclassMisa-additiveclassandGMD. InviewofTheorem1.2.1 M is a-algebra, ifM is a-system and in that case(G)M: Thus the theorem follows if we show thatM is a-system. GivenC X; denotebyD betheclassofallDX suchthatD\C 2 C M.19 CLAIM 1. If C 2M; thenD is a-additive class. C PROOF OF CLAIM 1. First X 2D since X\C =C 2M: Moreover, if C A;B2D andAB; thenA\C;B\C 2M and C (BnA)\C = (B\C)n(A\C)2M: Accordinglyfromthis,BnA2D :Finally,if(A ) isadisjointdenumer- C n n2N + ablecollectionofmembersofD , then (A \C) isdisjointdenumerable C n n2N + collection of members ofM and ( A )\C = (A \C)2M: n2N n n2N n + + Thus A 2D : n2N n C + CLAIM 2. If A2 G; thenMD : A PROOF OF CLAIM 2. If B 2 G; A\B 2 GM: Thus B 2 D : We A have proved thatGD and remembering thatM is the intersection of all A -additiveclassescontainingG Claim2followssinceD isa-additiveclass. A TocompletetheproofofTheorem1.2.2, observethatB2D ifandonly A if A2D : By Claim 2, ifA2G andB2M; thenB2D that isA2D : B A B ThusG D if B 2M. Now the de…nition ofM implies thatMD if B B B 2 M: The proof is almost …nished. In fact, if A;B 2 M then A 2 D B that isA\B2M: Theorem 1.2.2 now follows from Theorem 1.2.1. Theorem 1.2.3. Let  and  be positive measures on M = (G), where G is a -system, and suppose (A) = (A) for every A2G: (a) If  and  are probability measures, then  =: 1 (b) Suppose there exist E 2G; n2N ; such that X = E ; n + n n=120 E E :::; and 1 2 (E ) =(E )1; all n2N : n n + Then  =: PROOF. (a) Let D =fA2M; (A) = (A)g: It is immediate thatD is a -additive class and Theorem 1.2.2 implies that M =(G)D sinceGD andG is a-system. (b) If (E ) =(E ) = 0 for all all n2N , then n n + (X) = lim (E ) = 0 n n1 and, in a similar way, (X) = 0: Thus =: If (E ) =(E ) 0; set n n 1 1  (A) = (A\E ) and (A) = (A\E ) n n n n (E ) (E ) n n for eachA2M: By Part (a) = and we get n n (A\E ) =(A\E ) n n for eachA2M: Theorem 1.1.2(e) now proves that  =: Theorem 1.2.3 implies that there is at most one positive measure de…ned n onR such that the measure of any openn-cell inR equals its volume. n Next suppose f : X Y and let A  X and B  Y: The image of A and the inverse image of B are f(A) =fy; y =f(x) for somex2Ag and 1 f (B) =fx; f(x)2Bg

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