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A Note on Epi-Convergence

Published online by Cambridge University Press:  20 November 2018

Gerald Beer
Gerald Beer*
Affiliation:
Department of Mathematics, California State University Los Angeles, California 90032 U.S.A.
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Abstract

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Let LSC(X) denote the set of extended real valued lower semicontinuous functions on a metrizable space X. If f, f1, f2, f3 ,... is a sequence in LSC(X), we say 〈fn〉 is epi-convergent to f provided the sequence of epigraphs 〈epi fn 〉 is Kuratowski- Painlevé convergent to epi f. In this note we address the following question: what conditions on f and/or on X are necessary and sufficient for this mode of convergence to force epigraphical convergence with respect to the stronger Hausdorff metric and Vietoris topologies?

Keywords

Primary: 54B2026A15secondary: 54C35epi-convergencelower semicontinuous functionKuratowski-Painlevé convergenceFell topologyHausdorff distanceVietoris topology

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 3 , 01 September 1994, pp. 294 - 300
Copyright
Copyright © Canadian Mathematical Society 1994

References

[At] Attouch, H., Variational convergence for functions and operators, Pitman, New York, 1984.Google Scholar
[AF] Aubin, J.-P and Frankowska, H., Set-valued analysis, Birkhàuser, Boston, 1990.Google Scholar
[Bel] Beer, G., Upper semicontinuous functions and the Stone approximation theorem, J. Approx. Theory 34(1982), 1–11.Google Scholar
[Be2] Beer, G., Metric spaces with nice closed balls and distance functions for closed sets, Bull. Austral. Math. Soc. 35(1987), 81–96.Google Scholar
[CV] Castaing, C. and Valadier, M., Convex analysis and measurable multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977.Google Scholar
[DM] Dal Maso, G., An introduction to Γ-convergence, Birkhäuser, Boston, 1993.Google Scholar
[DG] Giorgi, E. D., Convergence problems for functionals and operators. In: Recent methods in nonlinear analysis, (eds. E. Giorgi, D., E. Magenes and U. Mosco), Pitagora Editrice, Bologna, 1979, 131–188.Google Scholar
[DSW] Dolecki, S., Salinetti, G. and Wets, R., Convergence of functions: equi-semicontinuity, Trans. Amer. Math. Soc. 276(1983), 409–429.Google Scholar
[Fe] Fell, J., A Hausdorff topology for the closed subsets of a locally compact non-Hausdorjf space, Proc. Amer. Math. Soc. 13(1962), 472–476.Google Scholar
[FLL] Francaviglia, S., Lechichki, A. and Levi, S., Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112(1985), 347–370.Google Scholar
[Ha] Hausdorff, F., Set theory, Chelsea Publishing Co., New York, New York, 1957.Google Scholar
[KT] Klein, E. and Thompson, A., Theory of correspondences, Wiley, New York, 1984.Google Scholar
[Ku] Kuratowski, K., Topology, Vol. 1, Academic Press, New York, 1966.Google Scholar
[Ma] Mazure, M.-L., Equi-semicontinuité inférieure, T-convergence, et convergence simple, Séminaire d'Analyse Convexe Montpellier 7(1981).Google Scholar
[Mi] Michael, E., Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71(1951), 152–182.Google Scholar
[No] Norberg, T., On Vervaat's sup vague topology, Ark. Mat. 28(1990), 139–144.Google Scholar
[Po] Poppe, H., Eine bemerkung Über trennungsaxiome im raum der abgeschlossenen teilmengen eines topologischen raumes, Arch. Math. 16(1965), 197–199.Google Scholar
[RW] Rockafellar, R. T. and Wets, R., Variational systems, an introduction. In: Multifunctions and integrands, (ed. G. Salinetti), Lecture Notes in Math. 1091, Springer-Verlag, Berlin, 1984, 1–54.Google Scholar
[SW] Salinetti, G. and Wets, R., On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60(1977), 211–226.Google Scholar
[Si] Sierpinski, W., Sur l'inversion du théorème de Bolzano-Weierstrass generalisé, Fund. Math. 34(1947), 155–156.Google Scholar
[Ve] Vervaat, W., Random upper semicontinuous functions and extremal processes, Report MS-8801,Center for Math, and Comp. Sci., Amsterdam, 1988.Google Scholar