[EN] We prove that, under some reasonable requirements, the unit balls of the spaces Lp(m) and Loo(m) of a vector measure of compact range m are compact with respect to the topology t_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 t_m are given.

P. Rueda acknowledges with thanks the support of the Ministerio de Economia y Competitividad (Spain) MTM2011-22417. E.A. Sanchez Perez acknowledges with thanks the support of the Ministerio de Economia y Competitividad (Spain) MTM2012-36740-C02-02.