Probability I Fall 2011 Contents 1 Measures 2 1.1 σ-fields and generators . 2 1.2 Outer measure . 4 1.3 Carath´eodory Theorem . 7 1.4 Product measure I . 8 1.5 Hausdorff measure and Hausdorff dimension . 10 2 Integrals 12 2.1 Measurable functions . 12 2.2 Monotone and bounded convergence . 14 2.3 Various modes of convergence . 14 2.4 Approximation by continuous functions . 15 2.5 Fubini and Radon-Nikodym Theorem . 17 3 Probability 18 3.1 Probabilistic terminology and notation . 18 3.2 Independence, Borel-Cantelli, 0-1 Law . 19 3.3 Lp-spaces . 21 3.4 Weak convergence of measures . 23 3.5 Measures on a metric space, tightness vs. compactness . 24 4 Appendix: Selected proofs 27 4.1 Section 1 . 27 4.2 Section 2 . 30 4.3 Section 3 . 33 1 1 Measures 1.1 σ-fields and generators A family F of subsets of a set S is called a field if (F1) ; 2 F and S 2 F; (F2) if A; B 2 F, then A [ B 2 F; (F3) if A 2 F, then Ac = S n A 2 F. Condition (F2), repeating itself, is equivalent to [n (F2f ) if A1;:::;An 2 F, then Ak 2 F. k=1 A field F is called a σ-field, if (F2f ) is strengthened to [1 (F2c) if A1;A2;::: 2 F, then Ak 2 F. k=1 In view of (3) and the De Morgan's laws, the union \[" in (F2) or (F2f ) or (F2c) can be replaced by the intersection \\". The members of a σ-field are called measurable sets. Proposition 1.1 The family 2S of all subsets of S is a σ-field. The intersection of an arbitrary collection of σ-fields is a σ-field. Proposition 1.2 Let G be any family of subsets of S. Then there is the unique smallest σ-field containing G. We denote this smallest σ-field by F = σ(G) and call G its generator. We say that G induces or generates or spans F. The Borel σ-field is generated by the topology in a topological space and its members are called Borel sets. In general, the definition is non-constructive although a simple construction exists for finite σ-fields, and some special countable σ-fields. Example 1.3 As a generator of the σ-field of Borel sets in Rn one may use the family of open balls, or the family of closed intervals n [a; b] = f x 2 R : ak ≤ xk ≤ bk; k = 1; : : : ; n g = [a1; b1] × · · · × [an; bn]; x or the family of half-open half-closed intervals n (a; b] = f x 2 R : ak < xk ≤ bk; k = 1; : : : ; n g : Since the Euclidean topology is countably generated, any of its generators will generate the Borel σ-field. 2 Example 1.4 The countable Cartesian product R1 = R × R × · · · is a metric space, e.g., under the metric X jx − y j ^ 1 d(x; y) = k k ; x = (x ); y = (y ); (1) 2k k k k Let (S; F) and (T; G) be measurable spaces. A function f : S ! T is called measurable with respect −1 −1 to (F; G) if f G ⊂ F, or equivalently, if f G0 ⊂ F for some generator G0 of G. In particular, for Borel measurable spaces spanned by topologies, every continuous function is Borel measurable. If T = Rn is equipped with the Borel σ-field, we talk about Borel measurable functions. Let (S; F) be a measurable space. A function µ : F! [0; 1], not constant 1, is said to be a measure, if µ is countably additive, i.e., [ X µ Ak = µAk; for every sequence of pairwise disjoint Ak 2 F: (2) k k The underlined restriction simply removes an unnecessary pathology. It is equivalent to the assumption µ; < 1. Then it follows that µ; = 0. F 1 The triple, (S; ; µ[) is called a measure space. A measure is called finite if µS < and a probability, if µS = 1. If S = Sk such that µSk < 1, then we call µ σ-finite. k Let S =6 ;, F = 2S. The measure concentrated at a 2 S, a.k.a the degenerated measure or the point mass, is defined as ( 1; if a 2 A; δaA = 1IA(a) = (3) 0; if a2 = A: ≥ 2 Let pk 0 and ak S. Then X µ = pk δak k P is called a discrete measure. It is a probability when k pk = 1 and the counting measure when pk = 1. 3 1.2 Outer measure Let S =6 ;. A set function ϕ : 2S ! [0; 1] is called an outer measure, or OM in short, if 1. '(;) = 0; 2. ϕ is monotonic, i.e., A ⊂ B ) '(A) ≤ '(B), for every A; B; ( [ ) X 3. ϕ is countably subadditive, i.e., ' Ak ≤ '(Ak), for every countable (Ak). k2K k2K Theorem 1.5 Let ' be an OM. Then the family def c M' = f A ⊂ S : for every P ⊂ A and Q ⊂ A ;'(P [ Q) = '(P ) + '(Q) g : (4) is a σ-field and ' is a measure on M'. The members of M' are called '-measurable sets. Intuitively, the restriction to measurable sets forces the additivity upon ϕ. The inequality \'(P [ Q) ≤ '(P ) + '(Q)" is contained in Condition 3 for OM. So, it suffices to consider c M' = f A ⊂ X : for every P ⊂ A and Q ⊂ A ;'(P [ Q) ≥ '(P ) + '(Q) g : A set function ' is called superadditive if '(A [ B) ≥ '(A) + '(B) when A \ B = ; A superadditive and monotonic ' is countably superadditive. For a topological space S, an OM is called a Borel outer measure (BOM in short) if M' contains Borel sets. In practice it suffices to show that M' contains a generator of Borel sets (e.g., every closed set is measurable). If (S; d) is a metric space, we call ' a Carath´eodory's OM outer measure (COM), if the additivity holds for metrically separated sets, i.e., d(A; B) > 0 ) '(A [ B) = '(A) + '(B); A; B ⊂ S: (5) Theorem 1.6 Let S be a metric space. Then COM ⊂ BOM. Some subsets D of S enjoy \natural" numerical values v(D) (\v" for \volume"), and the empty set most naturally should be assigned the value 0. Any such assignment gives rise to the OM and then by Theorem 1.5 to the true measure. Proposition 1.7 Let D ⊂ 2S contain ;, and v : D! [0; 1] be a set function such that v(;) = 0. The following formula defines an OM: ( ) X1 [ def 2 D ⊂ '(A) = 'v;D (A) = inf v(Dk): Dk ;A Dk : (6) k=1 k 4 The plethora of choices of the cover family D may lead to a redundancy or triviality. So we say that two 0 0 choices (D; v) and (D ; v ) are equivalent if they induce the same OM, i.e., (6) yields 'v;D = 'v0;D0 . Note the obvious implications: D ⊂ D0 ) ≥ 1. 'v;D 'v;D0 , ≤ 0 ) ≤ 2. v v 'v;D 'v0;D . In order to refine the former crude relation, we say that the cover D0 is finer than a cover D and write D ≺ D0, if [ X 8 8 2 D 9 f 0 g ⊂ D0 ⊂ 0 0 − ϵ > 0 D Dm D Dm and v(D) > v(Dm) ϵ. m m Proposition 1.8 D ≺ D0 ≥ If , then 'v;D 'v;D0 . If D ≺ D0 and D0 ≺ D, then the covers are called equivalent. Equivalent covers induce identical OM's and measures. Example 1.9 (Lebesgue measure) Let S = Rd and D consist of either closed, or open, or left-open right-closed intervals, or open or closed balls. Let v be the volume which is also called the length when d = 1 and the area when d = 2. All these families are equivalent. Example 1.10 (Lebesgue-Stieltjes measure) Let f : Rd ! R. d = 1. Assume that f is a nondecreasing right-continuous function. Let D = f (a; b] g and v(a; b] = f(b) − f(a). d ≥ 2. The d-dimensional increment can be defined by induction. Alternatively, we observe that Yn ( ) X 1I (x) = 1I −∞ (x ) − 1I −∞ (x ) = s(c) 1I −∞ (x ): (a;b] ( ;bj ] j ( ;aj ] j ( ;cj ] j j=1 c d d where 2 points c = (cj) 2 R have coordinates cj 2 f aj; bj g, and ( 1; if c uses an even number of coordinates of a s(c) = −1; if c uses an odd number of coordinates of a Then we put and require X def v(a; b] = s(c)f(c) ≥ 0; c2C for every a ≤ b, and that1 lim v(a; b] = 0 b&a For example, when d = 2, v((a1; a2); (b1; b2) ] = f(b1; b2) − f(a1; b2) − f(b1; a2) + f(a1; a2): 1The continuity condition is unnecessary for the purpose of mere construction of some measure. However, the type of the discontinuity of the underlying function should match the type of selected intervals, and a potential mismatch invariantly will entail unpleasant pathologies in later stages. 5 Abstract examples 1. Given a countable (finite) set N ⊂ S, one can put v(B) = card f N \ B g, which \counts" the number of points of the sequence that lie in B 2 F = 2S. Here, D = 2S. 2. Let H : S ! R be a function. For A 2 D = 2S put − v(A) = rangeA(H) = sup H(x) inf H(x): x2A x2A Let (S; d) be a metric space. 1. Define the diameter of a set as v(A) = diam(A) = sup d(x; y): x;y2A A modification of the metric may induce the same topology but the generated OM's could differ significantly.