1870 CHAPTER 59. BASIC PROBABILITY

and so

P(∩n−1

k=1A jk ∩BC) =n−1

∏k=1

P(A jk

)(1−P(B))

=n−1

∏k=1

P(A jk

)P(BC)

showing if B ∈ G(A j1 ··· ,A jn−1

), then so is BC. It is clear that G(A j1 ··· ,A jn−1

) is closed with

respect to disjoint unions also. Here is why. If{

B j}∞

j=1 are disjoint sets in G(A j1 ···A jn−1

),

P(∪∞

i=1Bi∩∩n−1k=1A jk

)=

∞

∑i=1

P(Bi∩∩n−1

k=1A jk

)=

∞

∑i=1

P(Bi)n−1

∏k=1

P(A jk

)=

n−1

∏k=1

P(A jk

) ∞

∑i=1

P(Bi)

=n−1

∏k=1

P(A jk

)P(∪∞

i=1Bi)

Therefore, by the π system lemma, Lemma 12.12.3 G(A j1 ···A jn−1

) = F jn . This proves the

induction step in going from r−1 to r.

What is a useful π system for B (E) , the Borel sets of E where E is a Banach space?

Recall the fundamental lemma used to prove the Pettis theorem. It was proved on Page645.

Lemma 59.4.2 Let E be a separable real Banach space. Sets of the form

{x ∈ E : x∗i (x)≤ α i, i = 1,2, · · · ,m}

where x∗i ∈ D′, a dense subspace of the unit ball of E ′ and α i ∈ [−∞,∞) are a π system,and denoting this π system by K , it follows σ (K ) = B (E). The sets of K are examplesof cylindrical sets. The D′ is that set for the proof of the Pettis theorem.

Proof: The sets described are obviously a π system. I want to show σ (K ) containsthe closed balls because then σ (K ) contains the open balls and hence the open sets andthe result will follow. Let D′ be described in Lemma 21.1.6. As pointed out earlier it can