Proceedings of the American Mathematical Society

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



A characterization of epi-convergence in terms of convergence of level sets

Authors: Gerald Beer, R. T. Rockafellar and Roger J.-B. Wets
Journal: Proc. Amer. Math. Soc. 116 (1992), 753-761
MSC: Primary 49J45; Secondary 47H19, 54B20, 54C30
MathSciNet review: 1119262
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Abstract: Let $ \operatorname{LSC} (X)$ denote the extended real-valued lower semicontinuous functions on a separable metrizable space $ X$. We show that a sequence $ \left\langle {{f_n}} \right\rangle $ in $ \operatorname{LSC} (X)$ is epi-convergent to $ f \in \operatorname{LSC} (X)$ if and only for each real $ \alpha $, the level set of height $ \alpha $ of $ f$ can be recovered as the Painlevé-Kuratowski limit of an appropriately chosen sequence of level sets of the $ {f_n}$ at heights $ {\alpha _n}$ approaching $ \alpha $. Assuming the continuum hypothesis, this result fails without separability. An analogous result holds for weakly lower semicontinuous functions defined on a separable Banach space with respect to Mosco epi-convergence.

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Keywords: Epi-convergence, level sets, lower semicontinuous function, Painlevé-Kuratowski convergence, Mosco convergence
Article copyright: © Copyright 1992 American Mathematical Society