Continuous version of the Choquet integral representation theorem

Volume 168 / 2005

Piotr Pucha/la Studia Mathematica 168 (2005), 15-24 MSC: 54C60, 54C65, 46A55, 46B22. DOI: 10.4064/sm168-1-2

Abstract

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.

Authors

  • Piotr Pucha/laInstitute of Mathematics and Computer Science
    Technical University of Cz/estochowa
    Dąbrowskiego 73
    42-200 Cz/estochowa, Poland
    e-mail

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