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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Spaces of vector measures


Author: A. Katsaras
Journal: Trans. Amer. Math. Soc. 206 (1975), 313-328
MSC: Primary 46E27
DOI: https://doi.org/10.1090/S0002-9947-1975-0365111-8
MathSciNet review: 0365111
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Abstract: Let $ {C_{rc}} = {C_{rc}}(X,E)$ denote the space of all continuous functions $ f$, from a completely regular Hausdorff space $ X$ into a locally convex space $ E$, for which $ f(X)$ is relatively compact. As it is shown in [8], the uniform dual $ {C'_{rc}}$ of $ {C_{rc}}$ can be identified with a space $ M(B,E')$ of $ E'$-valued measures defined on the algebra of subsets of $ X$ generated by the zero sets. In this paper the subspaces of all $ \sigma $-additive and all $ \tau $-additive members of $ M(B,E')$ are studied. Two locally convex topologies $ \beta $ and $ {\beta _1}$ are considered on $ {C_{rc}}$. They yield as dual spaces the spaces of all $ \tau $-additive and all $ \sigma $-additive members of $ M(B,E')$ respectively. In case $ E$ is a locally convex lattice, the $ \sigma $-additive and $ \tau $-additive members of $ M(B,E')$ correspond to the $ \sigma $-additive and $ \tau $-additive members of $ {C_{rc}}$ respectively.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0365111-8
Keywords: Locally convex spaces, strict topology, operator-valued measures, $ \sigma $-additive functionals, $ \tau $-additive functionals, locally convex lattice
Article copyright: © Copyright 1975 American Mathematical Society