Abstract:Criteria are given for determining the weak compactness, or otherwise, of the integration map associated with a vector measure. For instance, the space of integrable functions of a weakly compact integration map is necessarily normable for the mean convergence topology. Results are presented which relate weak compactness of the integration map with the property of being a bicontinuous isomorphism onto its range. Finally, a detailed description is given of the compactness properties for the integration maps of a class of measures taking their values in $\ell ^1$, equipped with various weak topologies.
Keywords: weakly compact integration map, factorization of a vector measure
AMS Subject Classification: Primary 46E30, 46A05; Secondary 47B07, 46G10