Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?

2011-05-24T15:52:22Z

Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with weak- $*$ topology (weak topology induced by the continuous functions). Consider a compact subset $S\subseteq \mathcal{P}(X)$ that is itself a Bauer simplex, i.e. it is convex, the set $\mathrm{ex}(S)$ of extreme points is compact, and the barycenter map (called resultant by Choquet), $r\colon \mathcal{P}\bigl(\mathrm{ex}(S)\bigr) \to S$ , $r(\mu)(A) = \int \nu(A)\, \mu(\mathrm{d}\nu)$ is injective (thus a homeomorphism).

In this situation, every continuous real valued function on $\mathrm{ex}(S)$ can be extended to a continuous affine function on $S$ .

Now my question is the following:

Can every continuous affine function $F\colon S\to \mathbb{R}$ on a Bauer simplex $S\subseteq\mathcal{P}(X)$ be extended to a continuous affine function on $\mathcal{P}(X)$ (and therefore $F(\mu) = \int f\,\mathrm{d}\mu$ for some continuous $f\colon X\to \mathbb{R}$ )?

I think the following question is equivalent: Can $F$ be extended to a continuous linear function on the vector space spanned by $S$ in the space $\mathcal{M}(X)$ of signed measures of bounded variation? (Then we can use Hahn Banach).

I am also interested in the following more general formulation. Every Bauer simplex $S$ is affinely homeomorphic to a probability simplex, namely $\mathcal{P}\bigl(\mathrm{ex}(S)\bigr)$ . If $S$ is given as a subset of a closed hyperplane (that does not contain $0$ ) of a locally convex topological vector space, can the affine homeomorphism be extended to a linear homeomorphism of the vector space spanned by $S$ into $\mathcal{M}\bigl(\mathrm{ex}(S)\bigr)$ ?

I guess this is extension cannot be done in general, but I do not know.

Any references, partial solutions, counter examples, and ideas are welcome.

Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions? - MathOverflow

Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?