Measure, Integration & Real Analysis, 2021a

398 Chapter 12 Probability Measures
EXERCISES 12
1 Suppose (Ω, F, P) is a probability space and A ∈F. Prove that A and Ω \ A
are independent if and only if P(A) =0 or P(A) =1.
2 Suppose P is Lebesgue measure on [0, 1]. Give an example of two disjoint Borel
subsets sets A and B of [0, 1] such that P(A) =P(B) = 1 2 , [0, 1 2
] and A are
independent, and [0, 1 2
] and B are independent.
3 Suppose (Ω, F, P) is a probability space and A, B ∈F. Prove that the following
are equivalent:
• A and B are independent events.
• A and Ω \ B are independent events.
• Ω \ A and B are independent events.
• Ω \ A and Ω \ B are independent events.
4 Suppose (Ω, F, P) is a probability space and {A k } k∈Γ is a family of events.
Prove the family {A k } k∈Γ is independent if and only if the family {Ω \ A k } k∈Γ
is independent.
5 Give an example of a probability space (Ω, F, P) and events A, B 1 , B 2 such
that A and B 1 are independent, A and B 2 are independent, but A and B 1 ∪ B 2
are not independent.
6 Give an example of a probability space (Ω, F, P) and events A 1 , A 2 , A 3 such
that A 1 and A 2 are independent, A 1 and A 3 are independent, and A 2 and A 3
are independent, but the family A 1 , A 2 , A 3 is not independent.
7 Suppose (Ω, F, P) is a probability space, A ∈F, and B 1 ⊂ B 2 ⊂···is an
increasing sequence of events such that A and B n are independent events for
each n ∈ Z + . Show that A and ⋃ ∞
n=1 B n are independent.
8 Suppose (Ω, F, P) is a probability space and {A t } t∈R is an independent family
of events such that P(A t ) < 1 for each t ∈ R. Prove that there exists a sequence
t 1 , t 2 ,...in R such that P ( ⋂ ∞n=1
A tn
) = 0.
9 Suppose (Ω, F, P) is a probability space and B 1 ,...,B n ∈Fare such that
P(B 1 ∩···∩B n ) > 0. Prove that
P(A ∩ B 1 ∩···∩B n )=P(B 1 ) · P B1 (B 2 ) ···P B1 ∩···∩B n−1
(B n ) · P B1 ∩···∩B n
(A)
for every event A ∈F.
10 Suppose (Ω, F, P) is a probability space and A ∈Fis an event such that
0 < P(A) < 1. Prove that
for every event B ∈F.
P(B) =P A (B) · P(A)+P Ω\A (B) · P(Ω \ A)
Measure, Integration & Real Analysis, by Sheldon Axler