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

Let LSC(X) denote the extended real-valued lower semicontinuous functions on a separable metrizable space X. We show that a sequence (fn) in LSC(X) is epi-convergent o f e LSC(X) if and only for each real a , the level set of height a of f can be recovered as the Painleve-Kuratowski limit of an appropriately chosen sequence of level sets of the fn at heights an approaching a. 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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