Compactness of integral operators and uniform integrability on measure spaces

Abstract

Let $(E,\mathcal{E},\mu )$ be a measure space and $G:E\times E\to [0,\infty ]$ 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 $\{G(x,\cdot )q:x\in E\}$ 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.