Compactness in $L^1$ of a vector measure
We study compactness and related topological properties in the space $L^1(m)$ 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 $L^1(m)$ 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 $L^1(m)$. The strong weakly compact generation of $L^1(m)$ is discussed as well.