A truncation criterion for compactness in asymptotic $L_p$ spaces

Abstract: 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. This gives a measure-theoretic counterpart to the Kolmogorov-Riesz theorem for asymptotic $L_p$ spaces on $\mathbb{R}^n$.