Compactness in $L^1$ of a vector measure
Volume 225 / 2014
Studia Mathematica 225 (2014), 259-282
MSC: 46E30, 46B50, 46G10.
DOI: 10.4064/sm225-3-6
Abstract
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.
