A different method for extending the representation theorems

Abstract

When we say that a probability measure μ on a compact convex set X “represents” a point x of X, we mean, of course, that μ(f) = f(x) for each continuous affine function f on X. One way of extending the representation theorems would be to show that this latter equality holds for a larger class of functions. For instance, Proposition 10.7 showed that it holds for upper semicontinuous (or lower semicontinuous) affine functions. In this section we will show that it holds for the affine functions of first Baire class, i.e., those affine functions which are the pointwise limit of a sequence of continuous (but not necessarily affine) functions on X.