The order convergence of martingales indexed by directed sets

Abstract

We obtain a condition on the underlying family of σ \sigma -algebras which is properly weaker than the Vitali property but which is also a sufficient condition for the order convergence of martingales of semibounded variation. We also obtain a sufficient condition for the order convergence of martingales of semibounded variation in terms of the finiteness of the extreme order limits of martingales of bounded variation.