Random Variables
In chapter 5 we considered random variables defined on a countable probability space (Ω, A , P). We now wish to consider an arbitrary abstract space, countable or not. If X maps Ω into a state space (F, F ), then what we will often want to compute is the
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In Chapter 5 we considered random variables defined on a countable probability space (Ω, A, P ). We now wish to consider an arbitrary abstract space, countable or not. If X maps Ω into a state space (F, F), then what we will often want to compute is the probability that X takes its values in a given subset of the state space. We take these subsets to be elements of the σ-algebra F of subsets of F . Thus, we will want to compute P ({ω : X(ω) ∈ A}) = P (X ∈ A) = P (X −1 (A)), which are three equivalent ways to write the same quantity. The third is enlightening: in order to compute P (X −1 (A)), we need X −1 (A) to be an element of A, the σ-algebra on Ω on which P is defined. This motivates the following definition. Definition 8.1. (a) Let (E, E) and (F, F) be two measurable spaces. A function X : E → F is called measurable (relative to E and F) if X −1 (Λ) ∈ E, for all Λ ∈ F. (One also writes X −1 (F) ⊂ E.) (b) When (E, E) = (Ω, A), a measurable function X is called a random variable (r.v.). (c) When F = R, we usually take F to be the Borel σ-algebra B of R. We will do this henceforth without special mention. Theorem 8.1. Let C be a class of subsets of F such that σ(C) = F. In order for a function X : E → F to be measurable (w.r.t. the σ-algebras E and F), it is necessary and sufficient that X −1 (C) ⊂ E. Proof. The necessity is clear, and we show sufficiency. That is, suppose that X −1 (C) ∈ E for all C ∈ C. We need to show X −1 (Λ) ∈ E for all Λ ∈ F. First note that X −1 (∪n Λn ) = ∪n X −1 (Λn ), X −1 (∩n Λn ) = ∩n X −1 (Λn ), and X −1 (Λc ) = (X −1 (Λ))c . Define B = {A ∈ F: X −1 (A) ∈ E}. Then C ⊂ B, and since X −1 commutes with countable intersections, countable unions, and complements, we have that B is also a σ-algebra. Therefore B ⊃ σ(C), and also F ⊃ B, and since F = σ(C) we conclude F = B, and thus X −1 (F) ⊂ σ(X −1 (C)) ⊂ E. We have seen that a probability measure P on R is characterized by the quantities P ((−∞, a]). Thus the distribution measure P X on R of a random variable X should be characterized by P X ((−∞, a]) = P (X ≤ a) and what is perhaps surprisingly nice is that being a random variable is J. Jacod et al., Probability Essentials © Springer-Verlag Berlin Heidelberg 2004
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8 Random Variables
further characterized only by events of the form {ω: X(ω) ≤ a} = {X ≤ a}. Indeed, what this amounts to is that a function is measurable — and hence a random variable — if and only if its distribution function is defined. Corollary 8.1. Let (F, F) = (R, B) and let (E, E) be an arbitrary measurable space. Let X, Xn be real-valued functions on E. a) X is measurable if and only if {X ≤ a} = {ω: X(ω) ≤ a} = X −1 ((−∞, a]) ∈ E, for each a; or iff {X < a} ∈ E, each a ∈ R. b) If Xn are measurable, sup Xn , inf Xn , lim supn→∞ Xn and lim inf n→∞ Xn are all measurable. c) If Xn are measurable and if Xn converges pointwise to X, then X is measurable. Proof. (a) From Theorem 2.1, we know for the Borel sets B on R that B = σ(C) where C = {(−∞, a]; a ∈ R). By hypothesis X −1 (C) ⊂ E, so (a) follows from Theorem 8.1. (b) Since
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