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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 ).