Borel selectors for upper semi-continuous set-valued maps. (English) Zbl 0588.54020

A set-valued map F from a topological space X to a topological space Y is said to be upper semi-continuous (u.s.c.) if the set \(\{\) x:F(x)\(\cap H\neq \emptyset \}\) is closed in X, whenever H is a closed set in Y. A function \(f: X\to Y\) is said to be a selector for F if f(x)\(\in F(x)\), for all \(x\in X\). The paper concentrates on the situation when X is metric and Y is a Banach space with its weak or weak-star topology, but some theorems have more general form.
In the beginning ”the nearest point selection” method allows the authors to obtain the theorem: Let X be a metric space and let Y be a Banach space with an equivalent strictly convex norm. Let F be a weakly u.s.c. set-valued map from X to Y, with non-empty convex and weakly compact values. Then F has a Borel measurable selector f of the first Borel class (i.e. \(f^{-1}(H)\) is a \(G_{\delta}\)-set in X, whenever H is a weak closed set in Y; the weak-star modification is also true).
But the main result of the paper deals with the Radon-Nikodym property (RNP) and culminates in the following: (i) Let X be a metric space and \(Y^*\) be the dual space to the Banach space Y. Suppose that \(Y^*\) has RNP. Let F be a weak-star u.s.c. set-valued map from X to \(Y^*\). Suppose further, that F takes only non-empty weak-star closed values. Then there exists a norm-Borel measurable selector f for F of the first Baire class (i.e. f is a pointwise limit of a sequence of norm continuous functions from X to \(Y^*)\); (ii) the conclusion in (i) holds for all such F only when \(Y^*\) has RNP. In the proof of (i) the RNP of \(Y^*\) is used in some equivalent form, namely for any bounded weak-star closed subset F of \(Y^*\) there is a point in which the identity map on F is weak-star-to-norm continuous. Some corollaries concerning maximal monotone maps, subdifferentials, attainment maps, and metric projections are also included.
A summary of above and related results has appeared [C. R. Acad. Sci., Paris, Sér. I 299, 125-128 (1984; Zbl 0569.28012)]. The papers announced there and connected with the present paper have already been published [the authors, R. W. Hansell and I. Labuda, Math. Z. 189, 297-318 (1985; Zbl 0544.54016); the first author, R. W. Hansell and M. Talagrand, J. Reine Angew. Math. 361, 201-220 (1985; Zbl 0573.54012)]. The reviewer does not know why almost all of these papers have been published in different journals.
Reviewer: B.Aniszczyk


54C65 Selections in general topology
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
Full Text: DOI


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