Continuous version of the Choquet integral representation theorem

Let E be a locally convex topological Hausdorff space, K a nonempty compact convex subset of E, mu a regular Borel probability measure on E and gamma > 0. We say that the measure mu gamma-represents a point x is an element of K if sup(parallel to f parallel to <= 1)\f (x) - integral(K) f d mu\ < gamma for any f is an element of E*. In this paper a continuous version of the Choquet theorem is proved, namely, 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). Two cases are considered: in the first case the values of P are compact, while in the second they are closed. For this purpose it is shown (using geometrical tools.) that the mapping t bar right arrow ext P(t) is lower semicontinuous. Continuous versions of the Krein-Milman theorem are obtained as corollaries.