1866 CHAPTER 59. BASIC PROBABILITY

Note that by Lemma 59.3.2 you can consider independent events in terms of indepen-dent σ algebras. That is, a set of independent events can always be considered as eventstaken from a set of independent σ algebras. This is a more general notion because here theσ algebras might have infinitely many sets in them.

Lemma 59.3.4 Suppose the set of random variables, {Xi}i∈I is independent. Also supposeI1 ⊆ I and j /∈ I1. Then the σ algebras σ (Xi : i ∈ I1) , σ (X j) are independent σ algebras.

Proof: Let B ∈ σ (X j) . I want to show that for any A ∈ σ (Xi : i ∈ I1) , it follows thatP(A∩B) = P(A)P(B) . Let K consist of finite intersections of sets of the form X−1

k (Bk)where Bk is a Borel set and k ∈ I1. Thus K is a π system and σ (K ) = σ (Xi : i ∈ I1) .Now if you have one of these sets of the form A = ∩m

k=1X−1k (Bk) where without loss of

generality, it can be assumed the k are distinct since X−1k (Bk)∩X−1

k

(B′k)= X−1

k

(Bk ∩B′k

),

then

P(A∩B) = P(∩m

k=1X−1k (Bk)∩B

)= P(B)

m

∏k=1

P(X−1

k (Bk))

= P(B)P(∩m

k=1X−1k (Bk)

).

Thus K is contained in

G ≡ {A ∈ σ (Xi : i ∈ I1) : P(A∩B) = P(A)P(B)} .

Now G is closed with respect to complements and countable disjoint unions. Here is why:If each Ai ∈ G and the Ai are disjoint,

P((∪∞i=1Ai)∩B) = P(∪∞

i=1 (Ai∩B))

= ∑i

P(Ai∩B) = ∑i

P(Ai)P(B)

= P(B)∑i

P(Ai) = P(B)P(∪∞i=1Ai)

If A ∈ G ,P(AC ∩B

)+P(A∩B) = P(B)

and so

P(AC ∩B

)= P(B)−P(A∩B)

= P(B)−P(A)P(B)

= P(B)(1−P(A)) = P(B)P(AC) .

Therefore, from the lemma on π systems, Lemma 12.12.3 on Page 329, it follows G ⊇σ (K ) = σ (Xi : i ∈ I1).

Lemma 59.3.5 If {Xk}rk=1 are independent random variables having values in Z a sep-

arable metric space, and if gk is a Borel measurable function, then {gk (Xk)}rk=1 is also