Convergence of Classes of Amarts Indexed by Directed Sets

Abstract

Let be a probability space, J a directed set filtering to the right. (X t) t∈J is a family of random variables adapted to an increasing family of σ-algebras . Vitali conditions V and V' on the σ-algebras, abstracting classical assumptions in Lebesgue's derivation theory, were made to insure essential convergence of martingales and submartingales (under proper boundedness assumptions). In reality these conditions, guaranteeing the existence of certain disjoint and properly measurable sets Bi, are better suited for study of amarts, since the sets Bi are a natural habitat and breeding ground for stopping times, thriving, as well known, precisely on disjoint and properly measurable sets. Thus K. Astbury [1] showed that the condition V, proved by K. Krickeberg to be sufficient for convergence of martingales (see [20] or Neveu [26], p. 98) is both necessary and sufficient for convergence of amarts. (We follow Neveu denoting by V the condition Krickeberg denotes by V+∞.) The Vitali condition V’, proved by Krickeberg [21] to be sufficient for convergence of submartingales, is shown here to be both necessary and sufficient for convergence of ordered amarts, defined similarly to amarts, except that the stopping times are ordered. We also introduce the controlled Vitali condition Ve, properly weaker than V’, and show that Ve is sufficient for convergence of controlled amarts, including submartingales.