Uniqueness of representing measures.

Abstract

The question of uniqueness of representing measures is a natural one, both in applications and in the theory itself. As always, one must specify clearly the context within which uniqueness is being asserted. What we would like most is a theorem which characterizes those compact convex X with the property that to each point there exists a unique measure that represents the point and is supported by the extreme points of X. Choquet has proved such a theorem for metrizable X, but there is no satisfactory result in the general case. On the other hand, Choquet and Meyer have characterized those X with the property that to each point there corresponds a unique maximal measure which represents the point. Since maximal measures are “supported” by the extreme points, it would seem that this answers the question, but the fact that “supported” is taken in an approximate sense makes a considerable difference. An example by Mokobodzki will show that uniqueness of maximal representing measures does not imply uniqueness of representing measures which vanish on Baire subsets of X ex X.