2004/05/31 by Piotr Puchała
Mathematics · #math.FA #msc:54C60 #msc:54C65 #msc:46A55 #msc:46B22
published as Studia Math. 168 (1), 2005, 15-24 · 9 pages, minor historical, editorial and bibliographical changes; version as appeared in the journal
arxiv created 2005/04/05 · arxiv updated 2009/12/01
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.