Facts: Let V and S be nonempty sets. 1) If C' is a collection of subsets of V then C Q 0(0) with equality if and only if C is a sigma algebra on V. 2) If C and D are collections of subsets of V that satisfy 0 Q D, then 0(0) Q 0(1)). 3) Let X : S t R be a function. For all sets C Q R dene the inverse image of C' with respect to the function X , denoted X_1(C), as the following set: X_1(C) : {w E 5': X0) E C} It can be shown that if A Q R then X_1(A}c = X_1(Ac}; If {Cil is a sequence of subsets of R then X_1(UE:10} = UE'Z1X_1(C') 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 (-oo, x] for some x E R. The set o (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:1 DIVA HP ( V ) I - X (1) We want to show that if X : S -> R is a Borel measurable function then (O) OB VA HP (V ) ,- X (2) In particular, (2) implies that if A E o(C) then {X ( A} is an event (and hence has a well defined probability P[X E A]). That is, a function X : S - R with the property that {X