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 | References | Similar Articles | Additional Information

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

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© Copyright 1975
American Mathematical Society