Continuous version of the Choquet integral representation theorem

Piotr Pucha/la¹

¹Institute of Mathematics and Computer Science Technical University of Cz/estochowa Dąbrowskiego 73 42-200 Cz/estochowa, Poland

Studia Mathematica, Tome 168 (2005) no. 1, pp. 15-24

Résumé

Let $E$ be a locally convex topological Hausdorff space, $K$ a nonempty compact convex subset of $E$, $\mu$ a regular Borel probability measure on $E$ and $\gamma >0$. We say that the measure $\mu$ $\gamma $-represents a point $x\in K$ if $\sup_{\| f\|\leq1}| f(x)-\int_{K}f\,d\mu | \gamma $ for any $f\in E^{\ast}$. In this paper a continuous version of the Choquet theorem is proved, namely, if $P$ is a continuous multivalued mapping from a metric space $T$ into the space of nonempty, bounded convex subsets of a Banach space $X$, then there exists a weak$^{\ast}$ continuous family $(\mu_{t})$ of regular Borel probability measures on $X$ $\gamma $-representing points in $P(t)$. Two cases are considered: in the first case the values of $P$ are compact, while in the second they are closed. For this purpose it is shown (using geometrical tools) that the mapping $t\mapsto \mathop{\rm ext}\nolimits P(t)$ is lower semicontinuous. Continuous versions of the Krein–Milman theorem are obtained as corollaries.

Détail

DOI : 10.4064/sm168-1-2

Keywords: locally convex topological hausdorff space nonempty compact convex subset regular borel probability measure gamma say measure gamma represents point sup leq int gamma ast paper continuous version choquet theorem proved namely continuous multivalued mapping metric space space nonempty bounded convex subsets banach space there exists weak ast continuous family regular borel probability measures gamma representing points cases considered first values compact while second closed purpose shown using geometrical tools mapping mapsto mathop ext nolimits lower semicontinuous continuous versions krein milman theorem obtained corollaries