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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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