1856 CHAPTER 59. BASIC PROBABILITY

Then S contains all the open sets and is clearly a σ algebra. Therefore, S contains theBorel sets. Let Bi be a Borel set in Ei. Then

n

∏i=1

Bi = ∩ni=1π

−1i (Bi) ,

a finite intersection of Borel sets.

Definition 59.1.4 A probability space is a measure space, (Ω,F ,P) where P is a measuresatisfying P(Ω) = 1. A random vector (variable) is a measurable function, X : Ω→ Zwhere Z is some topological space. It is often the case that Z will equal Rp. Assume Z is aseparable Banach space. Define the following σ algebra.

σ (X)≡{

X−1 (E) : E is Borel in Z}

Thus σ (X)⊆F . For E a Borel set in Z define

λ X (E)≡ P(X−1 (E)

).

This is called the distribution of the random variable, X. If∫Ω

|X(ω)|dP < ∞

then define

E (X)≡∫

Ω

XdP

where the integral is defined as the Bochner integral.

Recall the following fundamental result which was proved earlier but which I will givea short proof of now.

Proposition 59.1.5 Let (Ω,S ,µ) be a measure space and let X : Ω→ Z where Z is aseparable Banach space. Then X is strongly measurable if and only if X−1 (U) ∈S for allU open in Z.

Proof: To begin with, let D(a,r) be the closure of the open ball B(a,r). By Lemma21.1.6, there exists { fi} ⊆ B′, the unit ball in Z′ such that

∥z∥Z = supi{| fi (z)|}

ThenD(a,r) = {z : ∥a− z∥ ≤ r}= ∩i {z : | fi (z)− fi (a)| ≤ r}

= ∩i f−1i

(B( fi (a) ,r)

)It follows that

X−1 (D(a,r)) = ∩iX−1(

f−1i

(B( fi (a) ,r)

))= ∩i ( fi ◦X)−1

(B( fi (a) ,r)

)