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

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.