Continuous version of the Choquet integral representation theorem

• Autorzy: Piotr Puchała
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46B22: Radon-Nikod{\'y}m, Kre{\u\i}n-Milman and related properties
• 46A55: Convex sets in topological linear spaces; Choquet theory
• 54C60: Set-valued maps
• 54C65: Selections

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2005
• Tom: 168
• Numer: 1
• Strony: 15-24

Daty

wydano

2005

Twórcy

autor

Piotr Puchała

• Institute of Mathematics and Computer Science, Technical University of Częstochowa, Dąbrowskiego 73, 42-200 Częstochowa, Poland

Identyfikatory

DOI

10.4064/sm168-1-2