Measure, Integration & Real Analysis, 2021a

Chapter 12 Probability Measures 385
Compare the next result to the Borel–Cantelli Lemma (12.6).
12.10 relative of Borel–Cantelli Lemma
Suppose (Ω, F, P) is a probability space and {A n } n∈Z + is an independent family
of events such that ∑ ∞ n=1 P(A n)=∞. Then
Proof
P({ω ∈ Ω : ω ∈ A n for infinitely many n ∈ Z + })=1.
Let A = {ω ∈ Ω : ω ∈ A n for infinitely many n ∈ Z + }. Then
∞⋃ ∞⋂
12.11 Ω \ A = (Ω \ A n ).
m=1 n=m
If m, M ∈ Z + are such that m ≤ M, then
P ( ⋂ M (Ω \ A n ) ) M
= ∏ P(Ω \ A n )
n=m
n=m
12.12
=
M (
∏ 1 − P(An ) )
n=m
≤ e − ∑M n=m P(A n ) ,
where the first line holds because the family {Ω \ A n } n∈Z + is independent (see
Exercise 4) and the third line holds because 1 − t ≤ e −t for all t ≥ 0.
Because ∑ ∞ n=1 P(A n)=∞, by choosing M large we can make the right side of
12.12 as close to 0 as we wish. Thus
P ( ⋂ ∞ (Ω \ A n ) ) = 0
n=m
for all m ∈ Z + .Now12.11 implies that P(Ω \ A) =0. Thus we conclude that
P(A) =1, as desired.
For the rest of this chapter, assume that F = R. Thus, for example, if (Ω, F, P) is
a probability space, then L 1 (P) will always refer to the vector space of real-valued
F-measurable functions on Ω such that ∫ Ω
| f | dP < ∞.
12.13 Definition random variable; expectation; EX
Suppose (Ω, F, P) is a probability space.
• A random variable on (Ω, F) is a measurable function from Ω to R.
• If X ∈L 1 (P), then the expectation (sometimes called the expected value)
of the random variable X is denoted EX and is defined by
∫
EX = X dP.
Ω
Measure, Integration & Real Analysis, by Sheldon Axler