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 that always contain some elements that are barrelled and dense in , 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 -spaces defining the measure range space, the condition that they do not contain a copy of .

**[1]**J. Diestel, and B. Faires,*On vector measures*, Trans. Amer. Math. Soc.**198**(1974), 253-271. MR**0350420 (50:2912)****[2]**L. Drewnowski,*An extension of a theorem of Rosenthal on operators acting from*, Studia Math.**57**(1976), 209-215. MR**0423116 (54:11097)****[3]**J. C. Ferrando, and M. López-Pellicer,*Strong barrelledness properties in**and bounded finite additive measures*. Math. Ann.**287**(1990), 727-736. MR**1066827 (91i:46002)****[4]**M. Levin, and S. A. Saxon,*A note in the inheritance of properties of locally convex spaces by subspaces of countable codimension*, Proc. Amer. Math. Soc.**29**(1971), 97-102. MR**0280973 (43:6692)****[5]**P. Pérez Carreras, and J. Bonet*Barrelled locally convex spaces*, North Holland Math. Studies, vol. 131, Amsterdam, New York, Oxford, 1987. MR**880207 (88j:46003)****[6]**M. Valdivia,*On certain barrelled normed spaces*. Ann. Inst. Fourier (Grenoble)**29**(1979), 39-56. MR**552959 (81d:46006)****[7]**-,*Mackey convergence and the closed graph theorem*Arch. Math.**25**(1974), 649-656. MR**0374856 (51:11052)****[8]**-,*Sobre el teorema de la gráfica cerrada*. Collect. Math.**22**(1971), 51-72.

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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,
and -spaces,
barrelled spaces,
finitely (countably) additive vector measure,
bounded vector measure

Article copyright:
© Copyright 1992
American Mathematical Society