Deferred Epigraphical Convergence for Sequences of Lower Semicontinuous Functions

Abstract

In this paper, we introduce deferred epi-convergence for sequences of lower semicontinuous functions defined on metric spaces. The notion is formulated by using deferred Kuratowski convergence of epigraphs. In this framework, ordinary Ces\`aro averages are replaced by averages taken over the intervals $(p_n,q_n]$. We first define deferred lower and upper Kuratowski limits for sequences of closed sets. These set limits are then used to define the lower and upper deferred epi-limits of a sequence of functions. We prove that ordinary epi-convergence implies deferred epi-convergence and give an example showing that the converse need not hold. Finally, we obtain distance representations of the deferred epi-limit functions in terms of averaged distances to epigraphs.