Epi-convergence almost surely, in probability and in distribution

The paper deals with an epi-convergence of random real functions defined on a topological space. We follow the idea due to Vogel to split the epi-convergence into the lower semicontinuous approximation and the epi-upper approximation and localize them onto a given set. The approximations are shown to be connected to the miss (or hit) part of the ordinary Fell topology on sets. We introduce two procedures, called ‘localization’, separately for the miss-topology and the hit-topology on sets. Localization of the miss (or hit) part of the Fell topology on sets allows us to give a suggestion how to define the approximations in probability and in distribution. It is shown in the paper that in case of the finite-dimensional Euclidean space, the suggested approximations in probability coincide with the definition from Vogel and Lachout.