Foundations of Modern Probability Olav Kallenberg Springer [...]... class of all subsets of Ω Note that any - eld A is closed under monotone limits Thus, if A1 , A2 , ∈ A with An ↑ A or An ↓ A, then also A ∈ A A measurable space is a pair (Ω, A), where Ω is a space and A is a - eld in Ω 1 2 Foundations of Modern Probability For any class of - elds in Ω, the intersection (but usually not the union) is again a - eld If C is an arbitrary class of subsets of Ω,... µ), we define the µ-completion of A as the - eld Aµ = σ(A, Nµ ), where Nµ denotes the class of all subsets of µ-null sets in A The description of Aµ can be made more explicit, as follows Lemma 1.25 (completion) Consider a measure space (Ω, A, µ) and a Borel space (S, S) Then a function f : Ω → S is Aµ -measurable iff there exists some A-measurable function g satisfying f = g a.e µ Proof: With Nµ as before,... whenever f is the L2 -limit of a sequence in M Theorem 1.34 (orthogonal projection) Let M be a closed linear subspace of L2 Then any function f ∈ L2 has an a.e unique decomposition f = g + h with g ∈ M and h ⊥ M 18 Foundations of Modern Probability Proof: Fix any f ∈ L2 , and define d = inf{ f − g ; g ∈ M } Choose g1 , g2 , ∈ M with f − gn → d Using the linearity of M, the definition of d, and (10),... condition for changing the order of integration This requires a preliminary technical lemma 14 Foundations of Modern Probability Lemma 1.26 (sections) Fix two measurable spaces (S, S) and (T, T ), a measurable function f : S × T → R+ , and a - nite measure µ on S Then f (s, t) is S-measurable in s ∈ S for each t ∈ T , and the function t → µf (·, t) is T -measurable Proof: We may assume that µ is bounded... specified The elements of B(S) are called Borel sets In the case of the real line R, we shall often write B instead of B(R) More primitive classes than - elds often arise in applications A class C of subsets of some space Ω is called a π-system if it is closed under finite intersections, so that A, B ∈ C implies A ∩ B ∈ C Furthermore, a class D is a λ-system if it contains Ω and is closed under proper differences... measure space (S, S, µ) and a - eld T ⊂ S, let S µ denote the µ-completion of S, and let T µ be the - eld generated by T and the µ-null sets of S µ Show that A ∈ T µ iff there exist some B ∈ T and N ∈ S µ with A∆B ⊂ N and µN = 0 Also, show by an example that T µ may be strictly greater than the µ-completion of T 10 State Fubini’s theorem for the case where µ is any - nite measure and ν is the counting... existence; moments and continuity of paths Armed with the basic notions and results of measure theory from the previous chapter, we may now embark on our study of probability theory itself The dual purpose of this chapter is to introduce the basic terminology and notation and to prove some fundamental results, many of which are used throughout the remainder of this book In modern probability theory it is customary... independent random elements has probability zero or one As a consequence, any random variable that depends only on the “tail” of the sequence must be a.s constant This result and the related Hewitt–Savage zero–one law convey much of the flavor of modern probability: Although the individual elements of a random sequence are erratic and unpredictable, the long-term behavior may often conform to deterministic... and ν be probability kernels from S to T and from S × T to U , respectively, and consider two measurable functions f : S × T → R+ and g : S × T → U Then 20 Foundations of Modern Probability (i) µs f (s, ·) is a measurable function of s ∈ S; (ii) µs ◦ (g(s, ·))−1 is a kernel from S to U ; (iii) µ ⊗ ν is a kernel from S to T × U Proof: Assertion (i) is obvious when f is the indicator function of a set... there is a smallest - eld in Ω containing C, denoted by σ(C) and called the - eld generated or induced by C Note that σ(C) can be obtained as the intersection of all - elds in Ω that contain C A metric or topological space S will always be endowed with its Borel - eld B(S) generated by the topology (class of open subsets) in S unless a - eld is otherwise specified The elements of B(S) are called . A is a - eld in Ω. 1 2 Foundations of Modern Probability For any class of - elds in Ω, the intersection (but usually not the union) is again a - eld. If C is an arbitrary class of subsets of Ω,. tightness uniform topology on C(K, S) Skorohod’s J 1 -topology x Foundations of Modern Probability equicontinuity and tightness convergence of random measures superposition and thinning exchangeable. decomposition quasi–left-continuity compensation of random measures excessive and superharmonic functions additive functionals as compensators Riesz decomposition xii Foundations of Modern Probability 23.