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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Baire and $\sigma$-Borel characterizations of weakly compact sets in $M(T)$
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by T. V. Panchapagesan PDF
Trans. Amer. Math. Soc. 350 (1998), 4839-4847 Request permission


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
  • Email:
  • 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.
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 4839-4847
  • MSC (1991): Primary 28A33, 28C05, 28C15; Secondary 46E27
  • DOI:
  • MathSciNet review: 1615946