Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Convergence and approximation results for measurable multifunctions


Authors: R. Lucchetti, N. S. Papageorgiou and F. Patrone
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] H. Attouch, Variational convergence for functions and operators, Pitman, Boston, Mass., 1984. MR 773850 (86f:49002)
  • [2] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math., vol. 580, Springer, Berlin, 1977. MR 0467310 (57:7169)
  • [3] J. Diestel, Geometry of Banach spaces--Selected topics, Lecture Notes in Math., vol. 485, Springer, Berlin, 1975. MR 0461094 (57:1079)
  • [4] C. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72. MR 0367142 (51:3384)
  • [5] K, Kuratowski, Topology. I, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [6] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math. 3 (1969), 510-585. MR 0298508 (45:7560)
  • [7] G. Salinetti and R. Wets, On the convergence of closed valued measurable multifunctions, Trans. Amer. Math. Soc. 266 (1981), 275-289. MR 613796 (82k:28007)
  • [8] V. Toma, Quelgues problèmes de mésurabilité de multifonctions, Séminaire d'Analyse Convexe, Montpellier 13, 1983, pp. 6.1-6.17. MR 746115 (86a:28005)
  • [9] M. Tsukada, Convergence of best approximations in smooth Banach spaces, J. Approx. Theory 40 (1984), 301-309. MR 740641 (86a:41034)
  • [10] D. Wagner, Survey of measurable selection theorems, SIAM J. Control. Optim. 15 (1977), 859-903. MR 0486391 (58:6137)
  • [11] F. Hiai, Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. Amer. Math. Soc. 291 (1985), 613-627. MR 800254 (86k:60048)
  • [12] J. P. Aubin and A. Cellina, Differential inclusions, Springer, Berlin, 1984. MR 755330 (85j:49010)
  • [13] E. Klein and A. Thompson, Theory of correspondences, Wiley, New York, 1984. MR 752692 (86a:90012)
  • [14] M. A. Khan, On extensions of the Cournot-Nash theorem, Advances in Equilibrium Theory, (C. D. Aliprantis et al., eds.), Springer, New York, 1985. MR 873761 (88d:90129)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28A20, 54C60, 60G99

Retrieve articles in all journals with MSC: 28A20, 54C60, 60G99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0891162-4
Keywords: Kuratowski-Mosco convergence, measurable multifunction, Castaing representation, Aumann's selection theorem, complete $ \sigma $-field, countably simple multifunction, Trojanski's renorming theorem
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society