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Measure Theory and Probability Measure theory and probability Alexander Grigoryan University of Bielefeld Lecture Notes, October 2007 - February 2008 Contents 1 Construction of measures 3 1.1Introductionandexamples........................... 3 1.2 σ-additive measures ............................... 5 1.3 An example of using probability theory . .................. 7 1.4Extensionofmeasurefromsemi-ringtoaring................ 8 1.5 Extension of measure to a σ-algebra...................... 11 1.5.1 σ-rings and σ-algebras......................... 11 1.5.2 Outermeasure............................. 13 1.5.3 Symmetric difference.......................... 14 1.5.4 Measurable sets . ............................ 16 1.6 σ-finitemeasures................................ 20 1.7Nullsets..................................... 23 1.8 Lebesgue measure in Rn ............................ 25 1.8.1 Productmeasure............................ 25 1.8.2 Construction of measure in Rn. .................... 26 1.9 Probability spaces ................................ 28 1.10 Independence . ................................. 29 2 Integration 38 2.1 Measurable functions.............................. 38 2.2Sequencesofmeasurablefunctions....................... 42 2.3 The Lebesgue integral for finitemeasures................... 47 2.3.1 Simplefunctions............................ 47 2.3.2 Positivemeasurablefunctions..................... 49 2.3.3 Integrablefunctions........................... 52 2.4Integrationoversubsets............................ 56 2.5 The Lebesgue integral for σ-finitemeasure.................. 59 2.6 Convergence theorems............................. 61 2.7 Lebesgue function spaces Lp .......................... 68 2.7.1 The p-norm............................... 69 2.7.2 Spaces Lp ................................ 71 2.8ProductmeasuresandFubini’stheorem.................... 76 2.8.1 Productmeasure............................ 76 1 2.8.2 Cavalieriprinciple............................ 78 2.8.3 Fubini’stheorem............................ 83 3 Integration in Euclidean spaces and in probability spaces 86 3.1ChangeofvariablesinLebesgueintegral................... 86 3.2Randomvariablesandtheirdistributions...................100 3.3Functionalsofrandomvariables........................104 3.4Randomvectorsandjointdistributions....................106 3.5Independentrandomvariables.........................110 3.6Sequencesofrandomvariables.........................114 3.7Theweaklawoflargenumbers........................115 3.8Thestronglawoflargenumbers........................117 3.9 Extra material: the proof of the Weierstrass theorem using the weak law oflargenumbers................................121 2 1 Construction of measures 1.1 Introduction and examples Themainsubjectofthislecturecourseandthenotionofmeasure (Maß). The rigorous definition of measure will be given later, but now we can recall the familiar from the elementary mathematics notions, which are all particular cases of measure: 1. Length of intervals in R:ifI is a bounded interval with the endpoints a, b (that is, I is one of the intervals (a, b), [a, b], [a, b), (a, b]) then its length is defined by (I)= b a . | − | The useful property of the length is the additivity:ifanintervalI is a disjoint union of n a finite family Ik of intervals, that is, I = Ik,then { }k=1 k n F (I)= (Ik) . k=1 X N Indeed, let ai be the set of all distinct endpoints of the intervals I,I1,...,In enumer- { }i=0 ated in the increasing order. Then I has the endpoints a0,aN while each interval Ik has necessarily the endpoints ai,ai+1 for some i (indeed, if the endpoints of Ik are ai and aj with j>i+1 then the point ai+1 is an interior point of Ik, which means that Ik must in- tersect with some other interval Im). Conversely, any couple ai, ai+1 of consecutive points are the end points of some interval Ik (indeed, the interval (ai,ai+1) must be covered by some interval Ik; since the endpoints of Ik are consecutive numbers in the sequence aj , { } it follows that they are ai and ai+1). We conclude that N 1 n − (I)=aN a0 = (ai+1 ai)= (Ik) . − − i=0 k=1 X X 2. Area of domains in R2. The full notion of area will be constructed within the general measure theory later in this course. However, for rectangular domains the area is defined easily. A rectangle A in R2 is defined as the direct product of two intervals I,J from R: 2 A = I J = (x, y) R : x I,y J . × ∈ ∈ ∈ Then set © ª area (A)= (I) (J) . We claim that the area is also additive: if a rectangle A is a disjoint union of a finite family of rectangles A1,...,An,thatis,A = k Ak,then Fn area (A)= area (Ak) . k=1 X For simplicity, let us restrict the consideration to the case when all sides of all rectangles are semi-open intervals of the form [a, b).Considerfirst a particular case, when the 3 rectangles A1, ..., Ak form a regular tiling of A;thatis,letA = I J where I = Ii and × i J = Jj, and assume that all rectangles Ak have the form Ii Jj.Then j × F F area (A)= (I) (J)= (Ii) (Jj)= (Ii) (Jj)= area (Ak) . i j i,j k X X X X Now consider the general case when A is an arbitrary disjoint union of rectangles Ak. Let xi be the set of all X-coordinates of the endpoints of the rectangles Ak put in { } the increasing order, and yj be similarly the set of all the Y -coordinates, also in the increasing order. Consider{ the} rectangles Bij =[xi,xi+1) [yj,yj+1). × Then the family Bij forms a regular tiling of A and, by the first case, { }i,j area (A)= area (Bij) . i,j X On the other hand, each Ak is a disjoint union of some of Bij, and, moreover, those Bij that are subsets of Ak, form a regular tiling of Ak, which implies that area(Ak)= area (Bij) . Bij A X⊂ k Combining the previous two lines and using the fact that each Bij is a subset of exactly one set Ak,weobtain area (Ak)= area (Bij)= area (Bij)=area(A) . k k Bij A i,j X X X⊂ k X 3. Volume of domains in R3. The construction is similar to the area. Consider all boxes in R3, that is, the domains of the form A = I J K where I,J,K are intervals × × in R, and set vol (A)= (I) (J) (K) . Then volume is also an additive functional, which is proved in a similar way. Later on, we will give the detailed proof of a similar statement in an abstract setting. 4. Probability is another example of an additive functional. In probability theory, one considers a set Ω of elementary events, and certain subsets of Ω are called events (Ereignisse). For each event A Ω, one assigns the probability, which is denoted by ⊂ P (A) and which is a real number in [0, 1]. A reasonably defined probability must satisfy the additivity: if the event A is a disjoint union of a finite sequence of evens A1, ..., An then n P (A)= P (Ak) . k=1 X The fact that Ai and Aj are disjoint, when i = j, means that the events Ai and Aj cannot occur at the same time. 6 Thecommonfeatureofalltheaboveexampleisthefollowing.Wearegivenanon- empty set M (which in the above example was R, R2, R3, Ω), a family S of its subsets (the 4 families of intervals, rectangles, boxes, events), and a functional μ : S R+ := [0, + ) (length, area, volume, probability) with the following property: if A → S is a disjoint∞ n ∈ union of a finite family Ak of sets from S then { }k=1 n μ (A)= μ (Ak) . k=1 X A functional μ with this property is called a finitely additive measure. Hence, length, area, volume, probability are all finitely additive measures. 1.2 σ-additive measures As above, let M be an arbitrary non-empty set and S be a family of subsets of M. Definition. A functional μ : S R+ is called a σ-additive measure if whenever a set → N A S is a disjoint union of an at most countable sequence Ak k=1 (where N is either finite∈ or N = )then { } ∞ N μ (A)= μ (Ak) . k=1 X If N = then the above sum is understood as a series. If this property holds only for finite values∞ of N then μ is a finitely additive measure. Clearly, a σ-additive measure is also finitely additive (but the converse is not true). At first, the difference between finitely additive and σ-additive measures might look in- significant, but the σ-additivity provides much more possibilities for applications and is one of the central issues of the measure theory. On the other hand, it is normally more difficult to prove σ-additivity. Theorem 1.1 The length is a σ-additive measure on the family of all bounded intervals in R. Before we prove this theorem, consider a simpler property. N Definition. A functional μ : S R+ is called σ-subadditive if whenever A Ak → ⊂ k=1 where A and all Ak areelementsofS and N is either finite or infinite, then S N μ (A) μ (Ak) . ≤ k=1 X If this property holds only for finite values of N then μ is called finitely subadditive. Lemma 1.2 The length is σ-subadditive. Proof. Let I, Ik ∞ be intervals such that I ∞ Ik and let us prove that { }k=1 ⊂ k=1 ∞ S (I) (Ik) ≤ k=1 X 5 (the case N< follows from the case N = byaddingtheemptyinterval).Letusfix ∞ ∞ some ε>0 and choose a bounded closed interval I0 I such that ⊂ (I) (I0)+ε. ≤ For any k,chooseanopen interval I0 Ik such that k ⊃ ε (I0 ) (Ik)+ . k ≤ 2k Then the bounded closed interval I0 is covered by a sequence Ik0 of open intervals. By n{ } the Borel-Lebesgue lemma, there is a finite subfamily Ik0 that also covers I0.It j j=1 follows from the finite additivity of length that it is finitelyn subadditive,o that is, (I0) (Ik0 j ), ≤ j X (see Exercise 7), which implies that ∞ (I0) (I0 ) . ≤ k k=1 X This yields ∞ ε ∞ l (I) ε + (Ik)+ =2ε + (Ik) . ≤ 2k k=1 k=1 X ³ ´ X Since ε>0 is arbitrary, letting ε 0 we finish the proof. → Proof of Theorem 1.1. We need to prove that if I = k∞=1 Ik then ∞ F (I)= (Ik) k=1 X ByLemma1.2,wehaveimmediately ∞ (I) (Ik) ≤ k=1 X sowearelefttoprovetheoppositeinequality.Foranyfixed n N,wehave ∈ n I Ik.
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