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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
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 ${Cโ€™_{rc}}$ of ${C_{rc}}$ can be identified with a space $M(B,Eโ€™)$ of $Eโ€™$-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 $M(B,Eโ€™)$ 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 $M(B,Eโ€™)$ respectively. In case $E$ is a locally convex lattice, the $\sigma$-additive and $\tau$-additive members of $M(B,Eโ€™)$ correspond to the $\sigma$-additive and $\tau$-additive members of ${C_{rc}}$ respectively.

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Keywords: Locally convex spaces, strict topology, operator-valued measures, <IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img33.gif" ALT="$\sigma$">-additive functionals, <IMG WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\tau$">-additive functionals, locally convex lattice
Article copyright: © Copyright 1975 American Mathematical Society