A truncation criterion for compactness in asymptotic $L_p$ spaces

We prove a compactness criterion for asymptotic $L_p$ spaces over arbitrary measure spaces. Total boundedness is characterized by almost equiboundedness together with total boundedness in $L_p$ of all truncations. As a consequence, we obtain a simpler proof of the Kolmogorov–Riesz compactness theorem for asymptotic $L_p$ spaces on ${\mathbb{R}}^n$.