A truncation criterion for compactness in asymptotic $L_p$ spaces

TL;DR

This paper establishes a measure-theoretic compactness criterion for asymptotic Lp spaces, linking total boundedness with truncation properties, extending classical results like the Kolmogorov-Riesz theorem.

Contribution

It introduces a new truncation-based criterion for compactness in asymptotic Lp spaces over arbitrary measure spaces.

Findings

Characterizes total boundedness via almost equiboundedness and truncation in Lp.

Provides a measure-theoretic analogue of the Kolmogorov-Riesz theorem.

Extends classical compactness results to asymptotic Lp spaces on general measure 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 $\mathbb{R}^n$.