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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

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