Compactness in spaces of $p$-integrable functions with respect to a vector measure

Abstract

We prove that, under some reasonable requirements, the unit balls of the spaces $L^p(m)$ and $L^\infty(m)$ of a vector measure of compact range$m$ are compact with respect to the topology $\tau_m$ of pointwiseconvergence of the integrals. This result can be considered as a generalization of the classical Alaoglu Theorem to spaces of $p$-integrablefunctions with respect to vector measures with relatively compactrange. Some applications to the analysis of the Saks spaces defined by the norm topology and $\tau_m$ are given.