Compactness of averaging operators on Banach function spaces

Let $X$ be a Borel metric measure space such that each closed ball is of positive and finite measure. In this paper, we give a sufficient and necessary condition for averaging operators on a Banach function space $E(X)$ on $X$ to be compact. As a corollary, we show that the averaging operators on the Lorentz space $L^{p,q}(X)$ of $X$ is compact if and only if $X$ is bounded, in the case where $X$ is a doubling and Borel-regular metric measure space with some continuity between metric and measure.