Compactness in L1 of a vector measure

Abstract

We study compactness and related topological properties in the space L1(m) of a Banach space valued measure m when the natural topologies associated to convergence of vector valued integrals are considered. The resulting topological spaces are shown to be angelic and the relationship of compactness and equi-integrability is explored. A natural norming subset of the dual unit ball of L1(m) appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration operator is analyzed in relation with the positive Schur property of L1(m). The strong weaklycompact generation of L1(m) is discussed as well.