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

Beer, Gerald; Rockafellar, R. T.; Wets, Roger J.-B.

doi:10.1090/S0002-9939-1992-1119262-6

A characterization of epi-convergence in terms of convergence of level sets
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by Gerald Beer, R. T. Rockafellar and Roger J.-B. Wets

Proc. Amer. Math. Soc. 116 (1992), 753-761

DOI: https://doi.org/10.1090/S0002-9939-1992-1119262-6

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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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