The Radon-Nikodym-Property, σ-dentability and martingales in locally convex spaces.

Abstract

In this paper we give relations between the Radon Nikodym-Property (RNP), in sequentially complete locally convex spaces, mean convergence of martingales, and α-dentability. (RNP) is equivalent with the property that a certain class of martingales is mean convergent, while <7-dentability is equivalent with the property that the same class of martingales is mean Cauchy. We give an example of a σ-dentable space not having the (RNP). It is also an example of a sequentially incomplete space of in tegrable functions, the range space being sequentially complete.