Extremal integral representations

AbstractLet 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 ϵ 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