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 [f(n)] in LSC(X) is epi-convergent to f is-an-element-of LSC(X) if and only for each real alpha , the level set of height alpha of f can be recovered as the Painleve-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.