Convergence and approximation results for measurable multifunctions
HTML articles powered by AMS MathViewer
- by R. Lucchetti, N. S. Papageorgiou and F. Patrone PDF
- Proc. Amer. Math. Soc. 100 (1987), 551-556 Request permission
Abstract:
In this note we consider measurable multifunctions taking values in a separable Banach space. We show that if $F(\omega ) \subseteq {\text {s}} - \lim {F_n}(\omega )$, then any Castaing representation of the $F( \cdot )$ can be obtained as the strong limit of Castaing representations of the ${F_n}$. We also prove that any weakly measurable multifunction is the Kuratowski-Mosco limit of a sequence of countably simple multifunctions. Then we show that in reflexive Banach spaces this approximation property is equivalent to weak measurability. Finally we discuss the problem of measurability of the inferior and ${\text {w}}$-superior limits of a sequence of measurable multifunctions.References
- H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
- C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
- Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
- C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72. MR 367142, DOI 10.4064/fm-87-1-53-72
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- Umberto Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (1969), 510–585. MR 298508, DOI 10.1016/0001-8708(69)90009-7
- Gabriella Salinetti and Roger J.-B. Wets, On the convergence of closed-valued measurable multifunctions, Trans. Amer. Math. Soc. 266 (1981), no. 1, 275–289. MR 613796, DOI 10.1090/S0002-9947-1981-0613796-3
- VladimĂr Toma, Quelques problèmes de mesurabilitĂ© des multifonctions, SĂ©m. Anal. Convexe 13 (1983), exp. no. 6, 17 (French). MR 746115
- Makoto Tsukada, Convergence of best approximations in a smooth Banach space, J. Approx. Theory 40 (1984), no. 4, 301–309. MR 740641, DOI 10.1016/0021-9045(84)90003-0
- Daniel H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), no. 5, 859–903. MR 486391, DOI 10.1137/0315056
- Fumio Hiai, Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. Amer. Math. Soc. 291 (1985), no. 2, 613–627. MR 800254, DOI 10.1090/S0002-9947-1985-0800254-X
- Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330, DOI 10.1007/978-3-642-69512-4
- Erwin Klein and Anthony C. Thompson, Theory of correspondences, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1984. Including applications to mathematical economics; A Wiley-Interscience Publication. MR 752692
- M. Ali Khan, On extensions of the Cournot-Nash theorem, Advances in equilibrium theory (Indianapolis, Ind., 1984) Lecture Notes in Econom. and Math. Systems, vol. 244, Springer, Berlin, 1985, pp. 79–106. MR 873761, DOI 10.1007/978-3-642-51602-3_{5}
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 100 (1987), 551-556
- MSC: Primary 28A20; Secondary 54C60, 60G99
- DOI: https://doi.org/10.1090/S0002-9939-1987-0891162-4
- MathSciNet review: 891162