Abstract

Let ⟨X, d⟩ be a separable metric space. A set-valued function Γ: S ⇉ X defined on a measurable space S whose values are nonempty closed subsets of X is declared measurable provided for each open subset V of X, {s ∈ S: Γ(s) ∩ V ≠ ∅} is a measurable subset of S. In this paper, we look at the relationship between measurability so defined and the Borel measurability of Γ, viewed as a single-valued function into the nonempty closed subsets of X, equipped with either the Hausdorff metric topology or with the Attouch-Wets topology. Our analysis rests on cardinality arguments in conjunction with the representation of these hyperspaces as weak topologies. Applications are given to convex-valued multifunctions.

Author information

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Alberto Barbati

Dipartimento di Matematica, Università degli Studi di Milano 20133 Milano, Italy

Gerald Beer

Department of Mathematics, California State University, Los Angeles Los Angeles, CA 90032, USA

Christian Hess

CEREMADE, Université de Paris-Dauphine 75775 Paris 16, France

Keywords

  • Measurable multifunction
  • Hausdorff metric topology
  • Attouch-Wets topology
  • Effros sigma algebra
  • support functional

References

  1. H. Attouch, R. Lucchetti, R. Wets: The topology of the ρ-Hausdorff distance, Annali Mat. Pura. Appl. 160 (1991), 303–320.https://doi.org/10.1007/BF01764131Google Scholar
  2. H. Attouch, R. Wets: Quantitative stability of variational systems, I. The epigraphical distance. Trans. Amer. Math. Soc. 328 (1991), 695-730.https://doi.org/10.2307/2001800Google Scholar
  3. H. Attouch, R. Wets: Epigraphical processes: laws of large numbers for random lower semicontinuous functions, preprint, 1994.Google Scholar
  4. D. Azé, J.-P. Penot: Operations on convergent families of sets and functions, Optimization 21 (1990), 521–534.https://doi.org/10.1080/02331939008843576Google Scholar
  5. A. Barbati: A measurability criterion for closed, convex and non-empty valued multifunctions, preprint.Google Scholar
  6. G. Beer: Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), 117–126.https://doi.org/10.1090/S0002-9939-1990-0982400-8Google Scholar
  7. G. Beer: Topologies on closed and closed convex sets and the Effros measurability of set valued functions, Sém. d’Anal. Convexe Montpellier (1991), exposé N ◦ 2.Google Scholar
  8. G. Beer, A. Di Concilio: Uniform continuity on bounded sets and the Attouch-Wets topology, Proc. Amer. Math. Soc. 109 (1990), 427–436.https://doi.org/10.1090/S0002-9939-1990-1012924-9Google Scholar
  9. G. Beer, A. Lechicki, S. Levi, S. Naimpally: Distance functionals and the suprema of hyperspace topologies, Annali Mat. Pura Appl. 162 (1992), 367–381.https://doi.org/10.1007/BF01760016Google Scholar
  10. G. Beer, R. Lucchetti: The epi-distance topology: continuity and stability results with applications to convex optimization problems, Math. Operations Research 17 (1992), 715–726.https://doi.org/10.1287/moor.17.3.715Google Scholar
  11. G. Beer, R. Lucchetti: Weak topologies for the closed subsets of a metrizable space, Trans. Amer. Math. Soc. 335 (1993), 805–822.https://doi.org/10.1090/S0002-9947-1993-1094552-XGoogle Scholar
  12. G. Beer, R. Lucchetti: Well-posed optimization problems and a new topology for the closed subsets of a metric space, Rocky Mountain J. Math. 23 (1993), 1197–1220.https://doi.org/10.1216/rmjm/1181072488Google Scholar
  13. N. Bourbaki: Topologie générale, Chapter IX (2nd edition), Hermann, Paris, 1958.Google Scholar
  14. C. Castaing, M. Valadier: Convex analysis and measurable multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977.https://doi.org/10.1007/BFb0087685Google Scholar
  15. J.P.R. Christensen: Topology and Borel structure, North-Holland, Amsterdam, 1974.Google Scholar
  16. C. Constantini, S. Levi, J. Zieminska: Metrics that generate the same hyperspace convergence, Set-valued analysis 1 (1993), 141–158.https://doi.org/10.1007/BF01027689Google Scholar
  17. B. Cornet: Topologies sur les fermés d’un espace métrique, Cahiers de mathématiques de la décision 7309, Université de Paris Dauphine, 1973.Google Scholar
  18. N. Dunford, J. Schwartz: Linear operators, Part I, Wiley, New York, 1958.Google Scholar
  19. P. Enflo: A counterexample to the approximation problem in Banach spaces, Acta Mat. 130 (1973), 309–317.https://doi.org/10.1007/BF02392270Google Scholar
  20. R. Engelking: General topology, Heldermann Verlag, Berlin 1989.Google Scholar
  21. C. Hess: Quelques résultats sur la mesurabilité des multifunctions à valeurs dans un espace métrique séparable, Sém. d’Anal. Convexe Montpellier (1986), exposé N ◦ 1.Google Scholar
  22. C. Hess: Loi de probabilité des ensembles aléatoires à valeurs fermées dans un espace métrique séparable, C. R. Acad. Sci. Paris 296, série I (1983), 883–886.Google Scholar
  23. R. Holmes: A course in optimization and best approximation, Lecture Notes in Math. 257, Springer-Verlag, New York, 1972.https://doi.org/10.1007/BFb0059450Google Scholar
  24. E. Klein, A. Thompson: Theory of correspondences, Wiley, New York, 1984.Google Scholar
  25. K. Kuratowski: Topology, vol. 1, Academic Press, New York, 1966.https://doi.org/10.1016/B978-0-12-429201-7.50005-3Google Scholar
  26. A. Lechicki, S. Levi: Wijsman convergence in the hyperspace of a metric space, Boll. Un. Mat. Ital. 5-B (1987), 435–452.Google Scholar
  27. D. Walkup, R. Wets: Continuity of some convex-cone valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229–235.https://doi.org/10.1090/S0002-9939-1967-0209806-6Google Scholar

Suggested citation

A. Barbati, G. Beer, C. Hess. “The Hausdorff Metric Topology, the Attouch-Wets Topology, and the Measurability of Set-Valued Functions.” Journal of Convex Analysis 1 (1994), No. 1, 107–119. https://doi.org/10.68381/jca01008

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Published by Heldermann Verlag, 1994. Rights now held by Banach Press LLC.