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

Abstract

Asymptotic $L_p$-convergence, which resembles convergence in $L_p$, was introduced in \cite{alves2024mode}, motivated by a question in diffusive relaxation. The main purpose of this note is to compare asymptotic $L_p$-convergence with convergence in measure and in weak $L_p$ spaces. One of the results obtained provides a characterization of convergence in measure on finite measure spaces in terms of asymptotic $L_p$-convergence.