# A standard warm-up exercise Consider a probability space

**INSTRUCTIONS TO CANDIDATES**
###### ANSWER ALL QUESTIONS

1. (A standard warm-up exercise) Consider a probability space (Ω, A, P ) and assume that A, B ∈ A, and {An}n∈N ∈ A is a sequence of not necessarily disjoint sets. Based on the 3 Kolmogorov axioms for a probability measure (from the lecture notes), prove the following:

(a) P (B \ A) ≡ P (B ∩ Ac) = P (B) − P (B ∩ A)

(b) P (S∞n=1 An) ≤ Σ∞n=1 P (An)

(c) Bonferroni’s inequality for fixed n: P (Tn Ai) ≥ Σn P (Ai) − n + 1.

(d) Continuity: Suppose that for all ω ∈ Ω and some C ⊂ Ω

1C(ω) = lim

n→∞

1An (ω),

where 1C(ω) is the indicator function, which takes the value 1 if ω ∈ C and 0 if ω /∈ C.

i. Show that

C = [

\ Aj = \

[ Aj.

ii. Show that P ( j≥n Aj) ≤ P (C) ≤ P ( j≥n Aj).

iii. Based on these two steps conclude that C ∈ A and P (C) = limn→∞ P (An).

(e) Show that if if {Fn}n∈N is a sequence of σ-algebras, then n∞=1 Fn is a σ-algebra. (f) Consider the set Ω = {ω1, ω2, ω3} and the following two classes of subsets:

F1 = {∅, {ω1}, {ω2, ω3}, Ω} and F2 = {∅, {ω2}, {ω1, ω3}, Ω}.

i. Show that F1 and F2 are σ-algebras.

ii. Show that F1 ∩ F2 is a σ-algebra.

iii. Show that F1 ∪ F2 is not a σ-algebra.

iv. What does this tell you about unions of σ-algebras?

2. (Almost sure convergence) Let {Xn}n∈N ≡ {X1, X2, . . .} be a sequence of random variables and X be some random variable on some probability space (Ω, A , P ). For any ε > 0, we

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