Integrals of Set-Valued Functions with a Countable Range

Approximate versions, both of Lyapunov-type results on the compactness and convexity of the integral of a correspondence, and Fatou-type results on the preservation of upper semicontinuity by integration, are well known in the context of an infinite dimensional space. We report exact versions of these two types of results for integrals of Banach space valued correspondences with a countable range. We present results on both Bochner and Gel’fand integration.