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A note on a theorem of J. Diestel and B. Faires


Authors: J. C. Ferrando and M. López Pellicer
Journal: Proc. Amer. Math. Soc. 115 (1992), 1077-1081
MSC: Primary 46G10; Secondary 46A08
DOI: https://doi.org/10.1090/S0002-9939-1992-1091179-5
MathSciNet review: 1091179
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Abstract: Applying a property concerning certain coverings of $ l_0^\infty (X,\mathcal{A})$ that always contain some elements that are barrelled and dense in $ l_0^\infty (X,\mathcal{A})$, we generalize a localization theorem of M. Valdivia, relative to vector bounded finitely additive measures (Theorem 1), and obtain two different generalizations of a theorem of J. Diestel and B. Faires ensuring that certain finitely additive measures are countably additive (Theorems 2 and 3).

The original proof of the quoted theorem of Diestel and Faires uses a theorem of Rosenthal that is not required in our proof of Theorem 3. This avoids imposing over the Valdivia's $ {\Lambda _r}$-spaces defining the measure range space, the condition that they do not contain a copy of $ {l^\infty }$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1091179-5
Keywords: Closed graph theorems, dual locally complete spaces, $ {\Gamma _r}$ and $ {\Lambda _r}$-spaces, barrelled spaces, finitely (countably) additive vector measure, bounded vector measure
Article copyright: © Copyright 1992 American Mathematical Society