A NOTE ON A THEOREM OF DIESTEL,J. AND FAIRES,B.

Applying a property concerning certain coverings of l0 infinity (X, A) that always contain some elements that are barrelled and dense in l0 infinity (X, 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 AND(r)-spaces defining the measure range space, the condition that they do not contain a copy of l infinity.