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 Fréchet oscillation restriction condition and it is p-uniformly integrable. The proof is elementary.