Convergent processes, projective systems of measures and martingale decompositions

Abstract

The purpose of this paper is to show the equivalence of convergence, an associated projective system of measures and a martingale decomposition for a uniformly integrable stochastic process. Emphasis is placed on a direct juxtaposition of these concepts and on displaying underlying mechanisms.The impact of the martingale convergence theorem on contemporary probability theory has been immense. Therein lies the reason for numerous generalizations of both the basic martingale convergence theorem and the martingale concept itself.