Let E be a locally convex topological Hausdorff space, K a nonempty compact convex subset of E, μ a regular Borel probability measure on E and γ > 0. We say that the measure μ γ-represents a point x ∈ K if for any f ∈ 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 of regular Borel probability measures on X γ-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 ↦ ext P(t) is lower semicontinuous. Continuous versions of the Krein-Milman theorem are obtained as corollaries.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm168-1-2,
author = {Piotr Pucha\l a},
title = {Continuous version of the Choquet integral representation theorem},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {15-24},
zbl = {1077.46004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm168-1-2}
}
Piotr Puchała. Continuous version of the Choquet integral representation theorem. Studia Mathematica, Tome 166 (2005) pp. 15-24. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm168-1-2/