ON THE RADON-NiKODYM-PROPERTY AND MARTINGALE CONVERGENCE

It is well-known that there is an intimate connection between the RadonNikodym property and martingale convergence in a Banach space. This connection can be "localized" to a closed bounded convex subset of a Banach space. In this paper we are interested primarily in this connection for a bounded convex set which is not closed. If C is a bounded convex set in a locally convex space, C is said to have the martingale convergence ~ro~erty iff every martingale with values in C converges in measure. Since C is not assumed to be metrizable, it is appropriate to use martingales indexed by an arbitrary directed set~ and not restrict attention to sequential martingales. Similarly, C is said to have the Radon-Nikodym property iff every vector-valued measure defined on a probability space with average range in C has a derivative which has sufficiently strong measurability properties. The one-dimensional example of an open interval shows that the two properties are no longer equivalent. Theorem 2.4 describes the connection between the two notions. This paper is also concerned with an ordering on the tight probability measures on a bounded convex set C . The ordering ~ , which has been called "comparison of experiments", "the Choquet ordering"~ "the dilation ordering"~ and many others, can be described in many equivalent ways; they are given in Theorem 2.2. For example, means J J for all bounded continuous convex ctions f on C . Other descriptions of the ordering involve dilations and conditional expectations. Earlier versions of this theorem have been attributed to: Hardy 3 Littlewood and Polya~ Blackwell, Stein~ Sherman, Cartier and Strassen. One other result proved here deserves mention (Corollary 2.7). If C is a separable closed bounded convex subset of a Banach space, and if each point of C admits a unique representing measure on the extreme points of C ~ then C has the Radon-Nikodymproperty.