Webs and Bounded Finitely Additive Measures

Let Ms ms: sgS4 be a family of scalar bounded finitely additive measures defined on a s-algebra A. The Nikodym]Grothendieck boundedness theorem states that if M is simply bounded in A then M is uniformly bounded in A. In this paper we prove that if Vs An , n , . . . , n : p, n1, n2 . . . npgN4 is an increasing web 1 2 p in A, then there is a strand An n . . . n : igN4 such that if M is simply bounded in 1 2 i one An n . . . n then M is uniformly bounded in A Theorem 3.1.. This result is 1 2 i deduced from the fact that if Ws En n . . . n : p, n1, n2 , . . . , npgN4 is a linear 1 2 p increasing web in l` X, A., then there exists a strand E : igN4 such that 0 n1n2 . . . ni every E is barrelled and dense in l` X, A. Theorem 2.7.. From this strong n1n2 . . . ni 0 barrelledness condition previous results of the author jointly with J. C. Ferrando are improved here. These results are related to the classical result of Diestel and Faires in vector measures