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Transactions of the American Mathematical Society

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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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  • [BDS] R.G. Bartle, N. Dunford, and J.T. Schwartz, Weak compactness and vector measures, Canad. J.Math. 7, (1955), 289-305. MR 16:1123c
  • [DU] J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR 0453964
  • [Die] J. Dieudonné, Sur la convergence des suites de mesures de Radon, Anais Acad. Bras. Ciencias 23, (1951), 21-38. MR 13:121a
  • [Din] N. Dinculeanu, Vector Measures, Pergamon Press, New York, (1967).MR 34:6011
  • [DK] N. Dinculeanu and I. Kluvanek, On vector measures, Proc. London Math. Soc. (3) 17 (1967), 505–512. MR 0214722,
  • [DS] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
  • [E] R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
  • [G] A. Grothendieck, Sur les applications linéares faiblement compactes d'espaces du type C(K), Canad. J. Math. 5, (1953), 129-173. MR 15:438b
  • [H] P. R. Halmos, Measure Theory, Van Nostrand, New York, (1950). MR 11:504d
  • [K] Igor Kluvánek, Characterization of Fourier-Stieltjes transforms of vector and operator valued measures, Czechoslovak Math. J. 17 (92) (1967), 261–277 (English, with Russian summary). MR 0230872
  • [P1] T. V. Panchapagesan, On complex Radon measures. I, Czechoslovak Math. J. 42(117) (1992), no. 4, 599–612. MR 1182191
  • [P2] T. V. Panchapagesan, On complex Radon measures. II, Czechoslovak Math. J. 43(118) (1993), no. 1, 65–82. MR 1205231
  • [P3] -, Characterizations of weakly compact operators on $C_o(T)$, Trans. Amer. Math. Soc. 350 (1998), 4849-4867.
  • [T] Erik Thomas, L’intégration par rapport à une mesure de Radon vectorielle, Ann. Inst. Fourier (Grenoble) 20 (1970), no. 2, 55–191 (1971) (French, with English summary). MR 0463396

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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