Download 1 Event spaces and probability measures

Aalto University
Department of Mathematics and Systems Analysis
MS-E1600 - Probability theory, 2017/III
Problem set 1
K Kytölä & J Karjalainen
1 Event spaces and probability measures
Exercise session: Thu 5.1. at 14-16
Solutions due: Mon 9.1. at 10
1.1 Definition of a sigma-algebra. A collection F of subsets of a set Ω is a sigmaalgebra if it satisfies the following conditions:
(1) Ω ∈ F,
(1’) ∅ ∈ F,
(2) F ∈ F =⇒ F c ∈ F,
(3) F1 , F2 , · · · ∈ F =⇒ ∪∞
n=1 Fn ∈ F,
(3’) F1 , F2 , · · · ∈ F =⇒ ∩∞
n=1 Fn ∈ F.
Prove that for F to be a sigma-algebra, it is sufficient to assume that F satisfies
(1), (2), (3); that is, show that conditions (1), (2), (3) imply (10 ) and (30 ).
1.2 Sigma-algebras on small finite sets. Let a, b, c be three distinct points.
(a) Write down all sigma-algebras on Ω = {a, b}.
(b) Write down all sigma-algebras on Ω0 = {a, b, c}.
1.3 Probability distributions on countable spaces. Let Ω be a finite or a countably infinite
set, and denote by P(Ω) the collection of all subsets of Ω. A function
P p : Ω → R is
called a probability mass function (pmf ) if p(ω) ≥ 0 for all ω and ω∈Ω p(ω) = 1.
P
(a) Show that if p is a pmf on Ω, then the set function µ[E] = ω∈E p(ω) is a
probability measure on (Ω, P(Ω)).
(b) Show that if µ is a probability measure on (Ω, P(Ω)), then the function p(ω) =
µ[{ω}] is a pmf on Ω.
Continues on next page. . .
1.4 Monotone continuity of probability measures. For a sequence x1 , x2 , . . . of real numbers, we denote
• xn ↑ x if x1 ≤ x2 ≤ · · · and x = limn→∞ xn ,
• xn ↓ x if x1 ≥ x2 ≥ · · · and x = limn→∞ xn .
For a sequence F1 , F2 , . . . of sets we write
• Fn ↑ F if F1 ⊂ F2 ⊂ · · · and ∪n≥1 Fn = F .
• Fn ↓ F if F1 ⊃ F2 ⊃ · · · and ∩n≥1 Fn = F .
Suppose that (Ω, F, µ) is a probability space, that is, µ is a probability measure on
(Ω, F). Let F1 , F2 , . . . ∈ F be a sequence of events.
(a) Prove that Fn ↑ F =⇒ µ[Fn ] ↑ µ[F ].
(b) Prove that Fn ↓ F =⇒ µ[Fn ] ↓ µ[F ].
(c) Are the statements (a) and (b) still true in the case when µ is just assumed to
be a measure on (Ω, F), but not necessarily a probability measure?
1.5 Borel sets of the two-dimensional Euclidean space. The Borel sigma-algebra B(R2 )
is defined as the smallest sigma-algebra on R2 which contains all open sets in R2 .
Denote by
π(R2 ) = {(−∞, x] × (−∞, y] : x ∈ R, y ∈ R}.
the collection of closed lower rectangles in R2 . Prove that the collection of closed
lower rectangles generates B(R2 ), that is, show that B(R2 ) = σ(π(R2 )).
Hint: The same line of proof as in the one-dimensional case (e.g., [Jacod & Protter, Thm. 2.1] or [Williams, 1.2.a])
works here. You may use the fact that every open set in R2 can be written as a countable union ∪∞
n=1 Rn of open
rectangles of the form Rn = (an , bn ) × (a0n , b0n ).