Continuous version of the Choquet integral representation theorem

Piotr Puchała. "Continuous version of the Choquet integral representation theorem." Studia Mathematica 168.1 (2005): 15-24. <http://eudml.org/doc/284563>.

@article{PiotrPuchała2005,
abstract = {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 $sup_\{||f|| ≤ 1\} |f(x) - ∫_\{K\} fdμ| < γ$ 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 $(μ_\{t\})$ 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.},
author = {Piotr Puchała},
journal = {Studia Mathematica},
keywords = {Choquet representation theorem; continuous selection; extreme point; multivalued mapping; Radon-Nikodym property; strongly exposed point},
language = {eng},
number = {1},
pages = {15-24},
title = {Continuous version of the Choquet integral representation theorem},
url = {http://eudml.org/doc/284563},
volume = {168},
year = {2005},
}

TY - JOUR
AU - Piotr Puchała
TI - Continuous version of the Choquet integral representation theorem
JO - Studia Mathematica
PY - 2005
VL - 168
IS - 1
SP - 15
EP - 24
AB - 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 $sup_{||f|| ≤ 1} |f(x) - ∫_{K} fdμ| < γ$ 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 $(μ_{t})$ 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.
LA - eng
KW - Choquet representation theorem; continuous selection; extreme point; multivalued mapping; Radon-Nikodym property; strongly exposed point
UR - http://eudml.org/doc/284563
ER -