Extremal integral representations |
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Authors: | GA Edgar |
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Institution: | Department of Mathematics, Northwestern University, Evanston, Illinois 60201 U.S.A. |
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Abstract: | Let X be a closed bounded convex subset with the Radon-Nikodym property of a Banach space. For tight Borel probability measures μ, v on X, define μ ? v iff there is a dilation T on X such that T(μ) = v. Then, for every x?X, there is a measure μ on X which is maximal in the partial order ? and which has barycenter x. If X is separable, then μ(ex X) = 1 for all maximal measures μ. In general, a maximal measure need not be “on” ex X in this strong sense. If X is weakly compact, then a maximal measure is “on” ex X in the looser sense that μ(B) = 1 for all weak Baire sets B ? ex X. |
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