Kolmogorov$\unicode{x2013}$Riesz compactness in asymptotic $L_p$ spaces

Abstract

We extend the classical Kolmogorov-Riesz compactness theorem to the setting of asymptotic $L_p$ spaces on ${\mathbb{R}}^n$. These are nonlocally convex $\mathrm{F}$-spaces that contain the standard $L_p$ spaces as dense subspaces and include all measurable functions supported on sets of finite measure. In contrast with the classical $L_p$ setting, an additional almost equiboundedness condition is needed, and we prove that together with the natural tail and translation conditions it characterizes relative compactness. We conclude with illustrative examples.