Kolmogorov-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. As an application of our main result, we deduce a well-known characterization of relatively compact families of measurable functions in terms of almost equiboundedness and almost equicontinuity. We conclude with illustrative examples.