Relation between asymptotic $L_p$-convergence and some classical modes of convergence

TL;DR

This paper explores the relationship between asymptotic $L_p$-convergence and classical modes of convergence such as measure and weak $L_p$ convergence, providing characterizations and insights into their interconnections.

Contribution

It offers a new characterization of measure convergence using asymptotic $L_p$-convergence on finite measure spaces.

Findings

Asymptotic $L_p$-convergence is characterized in terms of measure convergence.

The paper establishes conditions under which asymptotic $L_p$-convergence aligns with convergence in measure.

Connections between asymptotic $L_p$-convergence and weak $L_p$ convergence are clarified.

Abstract

Asymptotic $L_p$-convergence, which resembles convergence in $L_p$, was introduced to address a question in diffusive relaxation. This note aims to compare asymptotic $L_p$-convergence with convergence in measure and in weak $L_p$ spaces. One of the results characterizes convergence in measure on finite measure spaces in terms of asymptotic $L_p$-convergence.