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Some results on density of extreme selections for measurable multifunctions. (English) Zbl 0612.28007
In this paper the author deals with measurable multifunctions F from A into $$2^ E$$ where (A,$${\mathcal A},m)$$ is a complete measure space with m nonnegative, finite and nonatomic, and where E is a locally convex Suslin space. Under different specific hypotheses on A and E the author establishes density properties of the set of measurable selections of F in the set of measurable selections of $$cl co F,$$ avoiding the assumptions of integrability on F and choosing suitable topologies in which the density is considered. The main results generalize and extend analogous theorems of Lyapunov’s type given by different authors in these last twenty years.
Reviewer: P.Pucci

##### MSC:
 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 54C65 Selections in general topology
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##### References:
 [1] Arkin, Dokl. Acad. Nauk SSSR. Tom 199 pp 1223– (1971) [2] Arkin, Russian Math. Surveys 27 pp 21– (1972) [3] Benamara, C. R. Acad. Sc. Paris 278 pp 1249– (1974) [4] Berliocchi, Bull. Soc. Math. France 101 pp 129– (1973) [5] Castaing, R. Acad. Sc. Paris 260 pp 3838– (1965) [6] Castaing, C. R. Acad. Sc. Paris 275 pp 1331– (1972) [7] , Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics. N. 580. Springer-Verlag, Berlin–Heidelberg–New York 1977 · Zbl 0346.46038 [8] , Analyse convexe et problèmes variationnels. Dunod–Gauthier–Villars (1974) · Zbl 0281.49001 [9] Generalized duality relations in extremal problems. Econ. i matem. metods 4.4 (1968) 597–610 (in Russian) [10] Ioffe, Surveys. 23 pp 6– (1968) [11] Kingman, J. London Math. Soc. 43 pp 2– (1968) [12] An approximation theorem for set-valued mappings. Acta Math. Vietnamica. Tl, N. 2 (1976) 97–104 · Zbl 0395.28006 [13] Densité des sélections extrémales d’une multifonction mesurable. Travaux du Séminaire d’Analyse Convexe. Montpellier 1979, N. 5 [14] Random Versions of Kakutani-Ky Fan’s Fixed Point Theorems. Journal of Math. Anal. and Appl. Vol 82 N. 2. (1981) 473–490 · Zbl 0487.60055 [15] Version vectorielle d’un théorème de densité et ses applications aux problèmes de contr??le C. R. Acad. Sc. Paris (in press) [16] Weak compactness of level sets in integral functionals. Troisieme Coll. d’Analyse fonctionneble integral functionals. Troisième Coll. d’Analyse fonctionneble (c. B. R. M. Liege, 1970). H. G. Garnir edit. [17] Uhl, Proc. Amer. Math. Soc. 23 pp 1– (1969) [18] Valadier, J. Math. Pures et Appl. 50 pp 265– (1971) [19] Semicontinuité de fonctionnelles intégrales. Séminaire d’Analyse convexe, Montpellier 1977, Exp. N. 2 [20] Quelques théorèmes bang bang. Séminaire d’Analyse convexe, Montpellier, 1981, Exp. N. 4 [21] Optimal control of Differential and Functional Equations. Academic Press, New York 1972
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