Let (S, F, P) be a probability triplet: S is the sample space; F is a...

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Let (S, F, P) be a probability triplet: S is the sample space; F is a sigma algebra on S; P : F → R is a probability measure. Define C as the set of all intervals in R of the type (-00, x] for some x e R. The set a(C) is called the Borel sigma algebra on R. A subset of R is said to be a Borel measurable set if it is in o(C). A function X : S →R is said to be a Borel measurable function if:' x-'(A) e F VA EC (1) We want to show that if X : S → R is a Borel measurable function then x-'(A) e F VA € o(C) (2) In particular, (2) implies that if A € o(C) then {X € A} is an event (and hence has a well defined probability P[X € A]). That is, a function X : S → R with the property that {X < x} is an event for all æ € R must also have the property that {X € A} is an event for all Borel measurable subsets AC R. a) Define H as the following set of subsets of R: H = {ACR:X-'(A) e F} Show that C C H. It immediately follows (by Fact 2) that o(C) C o(H). b) Argue that H is a sigma algebra on R. It immediately follows (by Fact 1) that o(H) = H. c) Argue that o(C) C H. Explain why this yields our desired conclusion (2). Let (S, F, P) be a probability triplet: S is the sample space; F is a sigma algebra on S; P : F → R is a probability measure. Define C as the set of all intervals in R of the type (-00, x] for some x e R. The set a(C) is called the Borel sigma algebra on R. A subset of R is said to be a Borel measurable set if it is in o(C). A function X : S →R is said to be a Borel measurable function if:' x-'(A) e F VA EC (1) We want to show that if X : S → R is a Borel measurable function then x-'(A) e F VA € o(C) (2) In particular, (2) implies that if A € o(C) then {X € A} is an event (and hence has a well defined probability P[X € A]). That is, a function X : S → R with the property that {X < x} is an event for all æ € R must also have the property that {X € A} is an event for all Borel measurable subsets AC R. a) Define H as the following set of subsets of R: H = {ACR:X-'(A) e F} Show that C C H. It immediately follows (by Fact 2) that o(C) C o(H). b) Argue that H is a sigma algebra on R. It immediately follows (by Fact 1) that o(H) = H. c) Argue that o(C) C H. Explain why this yields our desired conclusion (2).

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