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## Studia Mathematica

2005 | 168 | 1 | 15-24
Tytuł artykułu

### Continuous version of the Choquet integral representation theorem

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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.
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Kategorie tematyczne
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Tom
Numer
Strony
15-24
Opis fizyczny
Daty
wydano
2005
Twórcy
autor
• Institute of Mathematics and Computer Science, Technical University of Częstochowa, Dąbrowskiego 73, 42-200 Częstochowa, Poland
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