Abstract martingale convergence theorems

measure P is a completely additive non negative set function defined on a Borel field of sets, such that P{} = 1. We will be concerned with functions cc( ) defined on , and taking their values in a Banach space X. The sets of ^ will be referred to as the measurable sets. DEFINITION 1.1. x is a weak random variable if it is a weakly measurable function from to X. DEFINITION 1.2. x is a finitely {countably) valued random variable if it is constant on each of a finite (countable) number of disjunct measurable sets \ with = LMJ DEFINITION 1.3. # is a strong random variable if it is a strongly measurable function from to . DEFINITION 1.4. x is almost separably valued if there is a set A in ^// such that P{A} = 0 and x( -A) is separable. Note, x is strongly measurable if and only if it is weakly measurable and almost separably valued. (Pettis [15] and Hille-Phillips [9] Theorem 3.5.3, p. 72).