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Lattice copies of c₀ and in spaces of integrable functions for a vector measure

The spaces L¹(m) of all m-integrable (resp. L ¹ w ( m ) of all scalarly m-integrable) functions for a vector measure m, taking values in a complex locally convex Hausdorff space X (briefly, lcHs), are themselves lcHs for the mean convergence topology. Additionally, L ¹ w ( m ) is always a complex vector lattice; this is not necessarily so for L¹(m). To identify precisely when L¹(m) is also a complex vector lattice is one of our central aims. Whenever X is sequentially complete, then this is the case. If, additionally,...

Compactness in L¹ of a vector measure

J. M. CalabuigS. LajaraJ. RodríguezE. A. Sánchez-Pérez — 2014

Studia Mathematica

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...

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