1872 CHAPTER 59. BASIC PROBABILITY

Proof: First note that X−1 (σ (K )) is a σ algebra which contains X−1 (K ) and so itcontains σ

(X−1 (K )

). Thus

X−1 (σ (K ))⊇ σ(X−1 (K )

)Now let

G ≡{

A ∈ σ (K ) : X−1 (A) ∈ σ(X−1 (K )

)}Then G ⊇K . If A ∈ G , then

X−1 (A) ∈ σ(X−1 (K )

)and so

X−1 (A)C = X−1 (AC) ∈ σ(X−1 (K )

)because σ

(X−1 (K )

)is a σ algebra. Hence AC ∈ G . Finally suppose {Ai} is a sequence

of disjoint sets of G . Then

X−1 (∪∞i=1Ai) = ∪∞

i=1X−1 (Ai) ∈ σ(X−1 (K )

)again because σ

(X−1 (K )

)is a σ algebra. It follows from Lemma 12.12.3 on Page 329

that G ⊇ σ (K ) and this shows that whenever

A ∈ σ (K ) ,X−1 (A) ∈ σ(X−1 (K )

).

Thus X−1 (σ (K ))⊆ σ(X−1 (K )

).

With this lemma, here is the desired result about independent random variables. Essen-tially, you can reduce to the case of random vectors having values in Rn.

59.5 Reduction To Finite DimensionsLet E be a Banach space and let g ∈ (E ′)n . Then for x ∈ E, g◦x is the vector in Fn whichequals (g1 (x) ,g2 (x) , · · · ,gn (x)).

Theorem 59.5.1 Let Xi be a random variable having values in E a real separable Banachspace. The random variables {Xi}i∈I are independent if whenever

{i1, · · · , in} ⊆ I,

mi1 , · · · ,min are positive integers, and gmi1, · · · ,gmin

are respectively in

(M)mi1 , · · · ,(M)min

for M a dense subspace of E ′,{

gmi j◦Xi j

}n

j=1are independent random vectors having

values in Rmi1 , · · · ,Rmin respectively.