×

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

MSC:

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Asplund, E., Fréchet differentiability of convex functions.Acta Math., 121 (1968), 31–47. · Zbl 0162.17501
[2] Asplund, E. &Rockafellar, R. T., Gradients of convex functions.Trans. Amer. Math. Soc., 139 (1969), 443–467. · Zbl 0181.41901
[3] Bessaga, C. &Pelczynski, A.,Selected topics in infinite-dimensional topology. Polish Sci. Publishers, Warsaw, 1975. · Zbl 0304.57001
[4] Bourgain, J., On dentability and the Bishop-Phelps property.Israel J. Math., 28 (1977), 265–271. · Zbl 0365.46021
[5] Brezis, H.,Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Lecture Notes 5. North-Holland Publ. Co. Amsterdam, 1973.
[6] Browder, F. E., Nonlinear operators and nonlinear equations of evolution in Banach spaces.Proc. Sympos. Pure Math., vol. 18, part 2, Amer. Math. Soc., Providence, R.I., 1976. · Zbl 0327.47022
[7] Davis, W. J. &Phelps, R. R., The Radon-Nikodým property and dentable sets in Banach spaces.Proc. Amer. Math. Soc., 45 (1974), 119–122. · Zbl 0298.46046
[8] Diestel, J.,Geometry of Banach spaces-selected topics. Lecture Notes in Mathematics, 485, Springer-Verlag, New York, 1975. · Zbl 0307.46009
[9] Diestel, J. &Uhl, J. J.,Vector measures, Math. Surveys no. 15, Amer. Math. Soc., Providence, R.I., 1977. · Zbl 0369.46039
[10] Edgar, G. A. &Wheeler, R. F., Topological properties of Banach spaces.Pacific J. Math., 115 (1984), 317–350. · Zbl 0506.46007
[11] Godefroy, G.,Épluchabilité et unicité du prédual. Sem. Choquet, Comm. 11, 1977/78.
[12] Hansell, R. W., Borel measurable mappings for nonseparable metric spaces.Trans. Amer. Math. Soc., 161 (1971), 145–169. · Zbl 0232.28007
[13] –, On Borel mappings and Baire functions.Trans. Amer. Math. Soc., 194 (1974), 195–211 · Zbl 0295.54047
[14] Huff, F. E., Dentability and the Radon-Nikodým property.Duke Math. J., 41 (1974), 111–114. · Zbl 0285.46037
[15] James, R. C., Characterizations of reflexivity.Studia Math., 23 (1963/4), 205–216. · Zbl 0113.09303
[16] –, Weakly compact sets.Trans. Amer. Math. Soc., 113 (1964), 129–140. · Zbl 0129.07901
[17] –, A separable somewhat reflexive Banach space with non-separable dual.Bull. Amer. Math. Soc., 80 (1974), 738–743. · Zbl 0286.46018
[18] Jayne, J. E. &Rogers, C. A., Upper semi-continuous set-valued functions.Acta Math., 149 (1982), 87–125. · Zbl 0523.54013
[19] –, Borel selectors for upper semi-continuous multi-valued functions.J. Funct. Anal., 56 (1984), 279–299. · Zbl 0581.28007
[20] Kenderov, P. S., The set-valued monotone mappings are almost everywhere single-valued,C. R. Acad. Bulgare Sci., 27 (1974), 1173–1175. · Zbl 0339.47024
[21] –, Monotone operators in Asplund spaces.C. R. Acad. Bulgare Sci., 30 (1977), 963–964 · Zbl 0377.47036
[22] –, Dense strong continuity of ponntwise continuous mappings.Pacific J. Math., 89 (1980), 111–130. · Zbl 0458.54011
[23] Kuratowski, K.,Topology, vol. I. Academic Press, New York, 1966.
[24] Maynard, H. B., A geometrical characterization of Banach spaces having the Radon-Nikodým property.Trans. Amer. Math. Soc., 185 (1973), 493–500.
[25] Namioka, I. &Phelps, R. R., Banach spaces which are Asplund spaces.Duke Math. J., 42 (1975), 735–750. · Zbl 0332.46013
[26] Phelps, R. R., Dentability and extreme points in Banach spaces.J. Funct. Anal., 16 (1974), 78–90. · Zbl 0287.46026
[27] Phelps, R. R.,Differentiability of convex functions on Banach spaces. Lecture Notes, University College London, 1978. · Zbl 0396.46041
[28] Rieffel, M. A., Dentable subsets of Banach spaces, with applications to a Radon-Nikodým theorem, inFunctional Analysis (Proc. Conf., Irvine, Calif., 1966). B. R. Gelbaum, editor. Academic Press, pp. 71–77, 1967.
[29] Robert, R., Une généralisation aux opérateurs monotones des théorèmes de différentiabilité d’Asplund.C. R. Acad. Sci. Paris Sér. A, 278 (1974), 1189–1191.
[30] Rockafellar, R. T., Characterization of the subdifferentials of convex functions.Pacific J. Math., 17 (1966), 497–510. · Zbl 0145.15901
[31] –, Local boundedness of nonlinear monotone operators.Michigan Math. J., 16 (1969), 397–407. · Zbl 0184.17801
[32] –, On the maximal monotonicity of subdifferential mappings.Pacific J. Math., 33 (1970), 209–216. · Zbl 0199.47101
[33] Stegall, C., The Radon-Nikodým property in conjugate Banach spaces II.Trans. Amer. Math. Soc., 264 (1981), 507–519. · Zbl 0475.46016
[34] Bourgain, J. &Rosenthal, H., Geometrical implications of certain finite dimensional decompositions.Bull. Soc. Math. Belg., 32 (1980), 57–82. · Zbl 0463.46011
[35] Castaing, C. & Valadier, M.,Convex analysis and measurable multifunctions. Lecture Notes in Mathematics, 580, Springer-Verlag, 1977. · Zbl 0346.46038
[36] Dulst, v.D. &Namioka, I., A note on trees in conjugate Banach spaces.Indag. Math., 46 (1984), 7–10. · Zbl 0537.46025
[37] Jayne, J. E. &Rogers, C. A., Sélections borélinennes de multi-applications semi-continues supérieurment.C. R. Acad. Sci. Paris Sér. I, 299 (1984), 125–128.
[38] Kenderov, P. S., multivalued monotone mappings are almost everywhere single-valued.Studia Math., 56 (1976), 199–203. · Zbl 0341.47036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.