Epi-convergence in distribution of normal integrands with applications to sets of ϵ-optimal solutions

We derive necessary and sufficient conditions for epi-convergence in distribution of normal integrands.As a basic tool for the proof a new characterisation for distributional convergence of random closed sets is used.Our approach via the epi-topology allows us to show that, if a net of normal integrands epi-converges in distribution, then the pertaining sets of ϵ-optimal solutions converge in distribution in the underlying hyperspace endowed with the upper Fell topology.Under some boundedness and uniquenss assumptions the convergence even holds for the Fell topology.Finally, measurable selections converge weakly to a Choquet-capacity.