Compactness of integral operators and uniform integrability on measure spaces

Abstract

Let (E, E, µ) be a measure space and G : E × E → [0, ∞] be measurable. Moreover, let F ui denote the set of all q ∈ E + (measurable numerical functions q ≥ 0 on E) such that {G(x, •)q : x ∈ E} is uniformly integrable, and let F co denote the set of all q ∈ E + such that the mapping f → G(f q) := G(•, y)f (y)q(y) dµ(y) is a compact operator on the space E b of bounded measurable functions on E (equipped with the sup-norm).It is shown that F ui = F co provided both F ui and F co contain strictly positive functions.