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

DOI: http://dx.doi.org/10.12775/TMNA.2015.030

#### 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 pointwise convergence of the integrals. This result can be considered as a generalization of the classical Alaoglu Theorem to spaces of $p$-integrable functions with respect to vector measures with relatively compact range. Some applications to the analysis of the Saks spaces defined by the norm topology and $\tau_m$ are given.

#### Keywords

Banach function space; vector measure integration;compactness

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