Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T17:58:11Z http://mathoverflow.net/feeds/question/65867 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65867/are-affine-continuous-functions-on-bauer-sub-simplices-of-the-probability-measure Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions? Wolfgang Loehr 2011-05-24T15:52:22Z 2011-05-24T15:52:22Z <p>Let <code>$X$</code> be a <em>compact</em> (non-metrizable) <em>Hausdorff space</em> and <code>$\mathcal{P}(X)$</code> the set of <em>Radon probability measures</em> with weak-<code>$*$</code> topology (weak topology induced by the continuous functions). Consider a <em>compact subset</em> <code>$S\subseteq \mathcal{P}(X)$</code> that is itself a <strong>Bauer simplex</strong>, i.e. it is convex, the set <code>$\mathrm{ex}(S)$</code> of extreme points is compact, and the barycenter map (called resultant by Choquet), <code>$r\colon \mathcal{P}\bigl(\mathrm{ex}(S)\bigr) \to S$</code>, <code>$r(\mu)(A) = \int \nu(A)\, \mu(\mathrm{d}\nu)$</code> is injective (thus a homeomorphism).</p> <p>In this situation, every continuous real valued function on <code>$\mathrm{ex}(S)$</code> can be extended to a continuous affine function on <code>$S$</code>.</p> <hr> <p>Now my question is the following:</p> <blockquote> <p>Can every continuous affine function <code>$F\colon S\to \mathbb{R}$</code> on a Bauer simplex <code>$S\subseteq\mathcal{P}(X)$</code> be extended to a continuous affine function on <code>$\mathcal{P}(X)$</code> (and therefore <code>$F(\mu) = \int f\,\mathrm{d}\mu$</code> for some continuous <code>$f\colon X\to \mathbb{R}$</code>)?</p> </blockquote> <p>I think the following question is <strong>equivalent:</strong> Can <code>$F$</code> be extended to a continuous linear function on the vector space spanned by <code>$S$</code> in the space <code>$\mathcal{M}(X)$</code> of signed measures of bounded variation? (Then we can use Hahn Banach).</p> <hr> <p>I am also interested in the following <strong>more general</strong> formulation. Every Bauer simplex <code>$S$</code> is affinely homeomorphic to a probability simplex, namely <code>$\mathcal{P}\bigl(\mathrm{ex}(S)\bigr)$</code>. If <code>$S$</code> is given as a subset of a closed hyperplane (that does not contain <code>$0$</code>) of a locally convex topological vector space, can the affine homeomorphism be extended to a linear homeomorphism of the vector space spanned by <code>$S$</code> into <code>$\mathcal{M}\bigl(\mathrm{ex}(S)\bigr)$</code>?</p> <p>I guess this is extension cannot be done in general, but I do not know.</p> <p>Any references, partial solutions, counter examples, and ideas are welcome.</p>