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

TL;DR

This paper generalizes the Kolmogorov-Riesz compactness theorem to asymptotic $L_p$ spaces, revealing new conditions for compactness in these nonlocally convex spaces that extend classical results.

Contribution

It introduces a new characterization of relative compactness in asymptotic $L_p$ spaces, including an additional almost equiboundedness condition not present in classical $L_p$ spaces.

Findings

Additional almost equiboundedness condition is necessary.

Tail and translation conditions characterize compactness.

Includes illustrative examples demonstrating the theory.

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.