Martingale proofs of a general integral representation theorem

Summary

Abstract

Let K be a line-closed measure-convex bounded subset of a Banach space so that every relatively closed separable subset of K is analytic. By constructing certain martingales it is proved that if K has the Radon-Nikodým property, then for every k₀ in K there is a separable relatively closed convex subset K₀ of K and a Borel probability measure μ supported on the extreme points of L, for every relatively closed separable convex subset L of K with L ⊃ K₀, so that k₀ is the barycenter of μ; μ is uniquely determined by k₀ if and only if K is a simplex.

Introduction.

We give here a self-contained exposition of various known integral representation results, via the general theorem stated in the abstract. Both the existence and uniqueness parts of the theorem are obtained through the construction of certain martingales. Our main arguments and formulations are thus probabilistic in nature.

For the sake of orientation, we first recall the following representation result, due to E.G.F. Thomas [26].

Theorem.Let K be a closed bounded measure-convex Souslin subset of a locally convex space and assume that K has the Radon-Nikodým property (the RNP). Then for every x in K, there is a Borel probability measure μ supported on the extreme points of K so that x is the barycenter of μ; this μ is uniquely determined (for every given x) if and only if K is a simplex.