59.4. BANACH SPACE VALUED RANDOM VARIABLES 1871

be any dense subset of B′. Then

{x ∈ E : ||x−a|| ≤ r}

=

{x ∈ E : sup

f∈D′| f (x−a)| ≤ r

}

=

{x ∈ E : sup

f∈D′| f (x)− f (a)| ≤ r

}= ∩ f∈D′ {x ∈ E : f (a)− r ≤ f (x)≤ f (a)+ r}= ∩ f∈D′ {x ∈ E : f (x)≤ f (a)+ r and (− f )(x)≤ r− f (a)}

which equals a countable intersection of sets of the given π system. Therefore, every closedball is contained in σ (K ). It follows easily that every open ball is also contained in σ (K )because

B(a,r) = ∪∞n=1B

(a,r− 1

n

).

Since the Banach space is separable, it is completely separable and so every open set is thecountable union of balls. This shows the open sets are in σ (K ) and so σ (K ) ⊇B (E) .However, all the sets in the π system are closed hence Borel because they are inverse imagesof closed sets. Therefore, σ (K )⊆B (E) and so σ (K ) = B (E).

As mentioned above, we can replace D′ in the above with M, any dense subset of E ′.

Observation 59.4.3 Denote by Cα,n the set {β ∈ Rn : β i ≤ α i} . Also denote by gn an ele-ment of Mn with the understanding that gn : E→ Rn according to the rule

gn (x)≡ (g1 (x) , · · · ,gn (x)) .

Then the sets in the above lemma can be written as g−1n (Cα,n). In other words, sets of the

form g−1n (Cα,n) form a π system for B (E).

Next suppose you have some random variables having values in a separable Banachspace, E, {Xi}i∈I . How can you tell if they are independent? To show they are independent,you need to verify that

P(∩n

k=1X−1ik

(Fik

))=

n

∏k=1

P(

X−1ik

(Fik

))whenever the Fik are Borel sets in E. It is desirable to find a way to do this easily.

Lemma 59.4.4 Let K be a π system of sets of E, a separable real Banach space and let(Ω,F ,P) be a probability space and X : Ω→ E be a random variable. Then

X−1 (σ (K )) = σ(X−1 (K )

)