Remark on norm compactness in $L^p({\mu}, X)$

TL;DR

This paper establishes a new compactness criterion in vector-valued $L^p$ spaces, linking relative compactness to integrability, oscillation conditions, and uniform integrability, with an elementary proof.

Contribution

It provides a novel and elementary criterion for norm compactness in $L^p(\mu,X)$ spaces, connecting integral compactness, oscillation restrictions, and uniform integrability.

Findings

Characterizes relative norm compactness via integrals over measurable sets.

Introduces a Fréchet oscillation restriction condition.

Shows the criterion is elementary to prove.

Abstract

We prove a compactness criterion in $L^p({\mu},X)$: a subset of $L^p({\mu},X)$ is relatively norm compact iff the set of integrals of its functions over any measurable set is relatively norm compact, it satisfies the Fr\'echet oscillation restriction condition and it is p-uniformly integrable. The proof is elementary.