On the convergence of closedvalued measurable multifunctions
Authors:
Gabriella Salinetti and Roger J.B. Wets
Journal:
Trans. Amer. Math. Soc. 266 (1981), 275289
MSC:
Primary 28A20; Secondary 54C60
MathSciNet review:
613796
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Abstract: In this paper we study the convergence almost everywhere and in measure of sequences of closedvalued multifunctions. We first give a number of criteria for the convergence of sequences of closed subsets. These results are used to obtain various characterizations for the convergence of measurable multifunctions. In particular we are interested in the convergence properties of (measurable) selections.
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 C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math., vol. 580, SpringerVerlag, Berlin, 1977. MR 0467310 (57:7169)
 [2]
 R. T. Rockafeller, Integral functionals, normal integrands and measurable selections, Nonlinear Operators and Calculus of Variations, Lecture Notes in Math., vol. 543, SpringerVerlag, Berlin, 1977. MR 0512209 (58:23598)
 [3]
 G. Choquet, Convergences, Ann. Inst. Fourier (Grenoble) 23 (19471948), 55112. MR 0025716 (10:53d)
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 E. Michael, Topologies on space of subsets, Trans. Amer. Math. Soc. 71 (1951), 151182. MR 0042109 (13:54f)
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 D. Curtis and R. Schori, and homeomorphic to the Hilbert cube, Bull. Amer. Math. Soc. 80 (1974), 927931. MR 0353235 (50:5719)
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 G. Debreu, Integration of correspondences, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, Vol. II, Part 1, Univ. of California Press, Berkeley, Calif., 1966, pp. 351372. MR 0228252 (37:3835)
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 J. Neveu, Convergence presque sûre de martingales multivoques, Ann. Inst. H. Poincaré Sér. B 8 (1972), 17. MR 0331504 (48:9837)
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 G. Matheron, Random sets and integral geometry, Wiley, New York, 1975. MR 0385969 (52:6828)
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 G. Salinetti and R. Wets, On the convergence of sequences of convex sets in finite dimensions, SIAM Rev. 21 (1979), 1833. MR 516381 (80h:52007)
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 M. Sion, A theory of semigroup valued measures, Lecture Notes in Math., vol. 355, SpringerVerlag, Berlin, 1973. MR 0450503 (56:8797)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198106137963
PII:
S 00029947(1981)06137963
Keywords:
Measurable multifunction,
measurable selection,
convergence in probability,
convergence almost everywhere,
hyperspaces,
convergence in hyperspaces
Article copyright:
© Copyright 1981
American Mathematical Society
