Compactness of integral operators and uniform integrability on measure spaces

Abstract

Let $(E,\mathcal E,\mu)$ be a measure space and $G\colon E\times E\to [0,\infty]$ be measurable. Moreover, let $\mathcal F\!_{ui}$ denote the set of all $q\in\mathcal E^+$ (measurable numerical functions $q\ge 0$ on $E$) such that $\{G(x,\cdot)q\colon x\in E\}$ is uniformly integrable, and let $\mathcal F\!_{co}$ denote the set of all $q\in\mathcal E^+$ such that the mapping $f\mapsto G(fq) :=\int G(\cdot,y) f(y) q(y)\,d\mu(y)$ is a compact operator on the space $\mathcal E_b$ of bounded measurable functions on $E$ (equipped with the sup-norm). It is shown that $\mathcal F\!_{ui}=\mathcal F\!_{co} $ provided both $\mathcal F\!_{ui}$ and $\mathcal F\!_{co} $contain strictly positive functions.