728 CHAPTER 26. INDEPENDENCE
The Kolmogorov zero one law follows next. It has to do with something called a tailevent.
Definition 26.7.2 Let {Fn} be a sequence of σ algebras. Then Tn ≡ σ(∪∞
k=nFk)
where this means the smallest σ algebra which contains each Fk for k ≥ n. Then a tailevent is a set which is in the σ algebra, T ≡ ∩∞
n=1Tn.
As usual, (Ω,F ,P) is the underlying probability space such that all σ algebras arecontained in F .
Lemma 26.7.3 Suppose {Fn}∞
n=1 are independent σ algebras and suppose A is a tailevent and Aki ∈Fki , i = 1, · · · ,m are given sets. Then
P(Ak1 ∩·· ·∩Akm ∩A
)= P
(Ak1 ∩·· ·∩Akm
)P(A)
Proof: Let K be the π system consisting of finite intersections of the form
Bm1 ∩Bm2 ∩·· ·∩Bm j
where Bmi ∈Fki for ki > max{k1, · · · ,km} ≡ N. Thus σ (K ) = σ(∪∞
i=N+1Fi). Now let
G ≡{
B ∈ σ (K ) : P(Ak1 ∩·· ·∩Akm ∩B
)= P
(Ak1 ∩·· ·∩Akm
)P(B)
}Then clearly K ⊆ G . It is also true that G is closed with respect to complements andcountable disjoint unions. By the lemma on π systems, G = σ (K ) = σ
(∪∞
i=N+1Fi).
Since A is in σ(∪∞
i=N+1Fi)
due to the assumption that it is a tail event, it follows that
P(Ak1 ∩·· ·∩Akm ∩A
)= P
(Ak1 ∩·· ·∩Akm
)P(A) ■
Theorem 26.7.4 Suppose the σ algebras, {Fn}∞
n=1 are independent and supposeA is a tail event. Then P(A) either equals 0 or 1.
Proof: Let A ∈T ≡ ∩∞n=1Tn ≡ ∩∞
n=1σ(∪∞
k=nFk). I want to show that P(A) = P(A)2.
Since A is in T , it is in each σ(∪∞
k=nFk). Let K denote sets of the form Ak1 ∩ ·· · ∩Akm
for some m, Ak j ∈Fk j where each k j > n. Thus K is a π system and
σ (K ) = σ(∪∞
k=n+1Fk)≡Tn+1
LetG ≡
{B ∈Tn+1 ≡ σ
(∪∞
k=n+1Fk)
: P(A∩B) = P(A)P(B)}
Thus K ⊆ G because
P(Ak1 ∩·· ·∩Akm ∩A
)= P
(Ak1 ∩·· ·∩Akm
)P(A)
by Lemma 26.7.3. However, it is routine that G is closed with respect to countable disjointunions and complements. Therefore by the Lemma on π systems Lemma 9.3.2 on Page243, it follows G = σ (K ) = σ
(∪∞
k=n+1Fk).
Thus for any B ∈ σ(∪∞
k=n+1Fk)= Tn+1,P(A∩B) = P(A)P(B). However, A is in all
of these Tn+1 and so P(A∩A) = P(A) = P(A)2 so P(A) equals either 0 or 1. ■What sorts of things are tail events of independent σ algebras?