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

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 ${l^\infty }$.