Abstract integration of set-valued functions

We develop an abstract notion of integration for Effros measurable correspondences whose values are weakly compact subsets of a separable Banach space. This notion is built on a basic monotonicity hypothesis and the simple requirements that the integral assigns at most one value to any single-valued correspondence and evaluates the constant functions in the obvious way; linearity of the integral is not required. These hypotheses alone guarantee that the abstract integral is relatively weakly compact-valued, and its closed convex hull decomposes into the abstract integrals of the measurable selections from that correspondence. We use this decomposition theorem to prove a Fatou-type lemma and a monotone convergence theorem, and to derive necessary and sufficient conditions for the linearity and parametric continuity of the abstract integral. In turn, we apply our main results to obtain simple characterizations of some classical set-valued integrals, and derive (possibly nonadditive) aggregation methods for correspondences. All in all, we find that abstract integration theory yields many results about particular integrals for set-valued maps in a unified manner, often with minimal recourse to measure-theoretic arguments.