Applications of martingale system theorems

Basic definitions. The following definitions are all relative to a triple (Q, 13, Pr) in which Q is a space with points denoted by w, 13 is a Borel field of subsets of Q which includes Q, and Pr is a completely additive set function defined on sets of 13 such that for AC3B, 0<Pr A <Pr Q =1. A random variable is a function defined on Q having values on the real line extended by the adjunction of oo, + so and which is measurable with respect to 1B, i. e. , for every real number r, the X set x (X)? r is an element of 3. In the following the Borel fields discussed will be assumed to be subfields of 13 which include the set Q. If xt, tCT is any collection of random variables, the Borel field generated by this collection is the smallest Borel field with respect to which every member of the collection is measurable. A stochastic process is a collection xt, Ft, tC T, where T is a subset of the extended real line, Ft, t CT is an increasing collection of Borel fields in the sense that for t1 <t2, Ft1CFt2, and xt is a random variable measurable with respect to Ft or equal for almost all X to such a function. If the collection Xt, tCT is referred to as a stochastic process, it will be understood that Ft is the Borel field generated by x8, s<t, sET. By means of the probability measure Pr we define a Lebesgue integral and if a random variable x is integrable, we say that the expected value of x exists and define the expected value, written Ex, by Ex =fsxdpr. We do not require that E x be finite. Let x be a random variable such that Ex exists and Fbe a Borel field. Then4 defined bykA =fAxdpr forAE F is a completely additive set function defined on F which is absolutely continuous with respect to probability measure, in the sense that C A = 0 if Pr A =0. By a generalization of the Radon-NikodYm theorem 5, p. 169 there exists a random variable y which is measurable with respect to F, for which E y exists, and such that 0 A =fAydpr. The random variable y is unique up to a set of probability measure zero. This discussion justifies the following definition. DEFINITION 1. 1. Let x be a random variable such that Ex exists and let F be a Borel field. The conditional expectation of x relative to F, written Ex||F, is defined as any o function y which is equal for almost all X to a