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Continuous version of the Choquet integral representation theorem. (English) Zbl 1077.46004
The classical Minkowski-Carathéodory representation theorem states that each point of a compact convex set $$K$$ in $$\mathbb R^n$$ can be written as a convex combination of at most $$n+1$$ extreme points of $$K$$. This theorem was generalized by G. Choquet [Semin. Bourbaki 9 (1956/57), No. 139 (1959; Zbl 0121.09204)] who proved that each point $$k$$ of a compact, convex metrizable subset $$K$$ of a locally convex Hausdorff topological space $$X$$ is a barycenter of a regular Borel probability measure $$\mu$$ on $$X$$, supported by the extreme points of $$K$$, i.e., $$f(x)=\int_K f d\mu$$, for any $$f\in E^{*}$$, where $$\mu(\text{ext}\, K)=1$$ and ext $$K$$ stands for the set of extreme points of $$K$$.
In the present, paper the author shows that an analogue of the Choquet theorem holds for “moving” sets, which are values of a multivalued mapping from a metric space $$T$$ into suitable subsets of a Banach space $$X$$. More precisely: let $$E$$ be a locally convex topological Hausdorff space, $$K$$ a nonempty compact subset of $$E$$, $$\mu$$ a regular Borel probability measure on $$E$$ and $$\gamma >0$$. The measure $$\mu$$ is said to $$\gamma$$-represent a point $$x\in K$$ if $$\sup_{\| f\| \leq 1}| f(x)-\int_K f \,d\mu | <\gamma$$ for any $$f\in E^{*}$$. The author proves that 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* continuous family $$(\mu_t)$$ of regular Borel probability measures on $$X$$ $$\gamma$$-representing points in $$P(t)$$. Continuous versions of the Krein-Milman theorem are obtained as corollaries.
##### MSC:
 46A55 Convex sets in topological linear spaces; Choquet theory 54C60 Set-valued maps in general topology 54C65 Selections in general topology 46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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