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

Abstract

Let LSC ⁡ ( X ) \operatorname {LSC} (X) denote the extended real-valued lower semicontinuous functions on a separable metrizable space X X . We show that a sequence ⟨ f n ⟩ \left \langle {{f_n}} \right \rangle in LSC ⁡ ( X ) \operatorname {LSC} (X) is epi-convergent to f ∈ LSC ⁡ ( X ) f \in \operatorname {LSC} (X) if and only for each real α \alpha , the level set of height α \alpha of f f can be recovered as the Painlevé-Kuratowski limit of an appropriately chosen sequence of level sets of the f n {f_n} at heights α n {\alpha _n} approaching α \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.