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Baire and $\sigma$-Borel characterizations of weakly compact sets in $M(T)$

Author: T. V. Panchapagesan
Journal: Trans. Amer. Math. Soc. 350 (1998), 4839-4847
MSC (1991): Primary 28A33, 28C05, 28C15; Secondary 46E27
MathSciNet review: 1615946
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Abstract: Let $T$ be a locally compact Hausdorff space and let $M(T)$ be the Banach space of all bounded complex Radon measures on $T$. Let $\mathcal {B}_o(T)$ and $\mathcal {B}_c(T)$ be the $\sigma$-rings generated by the compact $G_\delta$ subsets and by the compact subsets of $T$, respectively. The members of $\mathcal {B}_o(T)$ are called Baire sets of $T$ and those of $\mathcal {B}_c(T)$ are called $\sigma$-Borel sets of $T$ (since they are precisely the $\sigma$-bounded Borel sets of $T$). Identifying $M(T)$ with the Banach space of all Borel regular complex measures on $T$, in this note we characterize weakly compact subsets $A$ of $M(T)$ in terms of the Baire and $\sigma$-Borel restrictions of the members of $A$. These characterizations permit us to give a generalization of a theorem of Dieudonné which is stronger and more natural than that given by Grothendieck.

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Additional Information

T. V. Panchapagesan
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Los Andes, Mérida, Venezuela

Keywords: Bounded complex Radon measures, uniform $\sigma$-additivity, uniform Baire inner regularity, uniform $\sigma$-Borel inner regularity, uniform Borel inner regularity, weakly compact sets
Received by editor(s): November 17, 1995
Additional Notes: Supported by the C.D.C.H.T. project C-586 of the Universidad de los Andes, Mérida, and by the international cooperation project between CONICIT-Venezuela and CNR-Italy.
Article copyright: © Copyright 1998 American Mathematical Society