This is an announcement for the paper "Continuous version of the Choquet Integral Reperesentation Theorem" by Piotr Puchala.

Abstract: The Choquet - Bishop - de Leeuw theorem states that each element of a compact convex subset of a locally convex topological Hausdorff space is a barycenter of a probability measure supported by the set of extreme points of that set. By the Edgar - Mankiewicz result this remains true for nonempty closed bounded and convex set provided it has Radon - Nikodym property. In the paper it is shown, that Choquet - type theorem holds also for "moving" sets: they are values of a certain multifunction. Namely, the existence of a suitable weak* continuous family of probability measures "almost representing" points of such sets is proven. Both compact and noncompact cases are considered. The continuous versions of the Krein - Milman theorem are obtained as corollaries.

Archive classification: Functional Analysis

Mathematics Subject Classification: 54C60; 54C65; 46A55; 46B22

Citation: Studia Math. 168 (1), 2005, 15-24

Remarks: 9 pages, minor historical, editorial and bibliographical changes;

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http://arXiv.org/abs/math.FA/0405217

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