Webs and bounded finitely additive measures

[摘要] Let M = {mu s: s is an element of S} be a family of scalar bounded finitely additive measures defined on a sigma-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 V = {A(n1,n2,...,np): p, n(1), n(2) ... n(p) is an element of N} is an increasing web in A, then there is a strand {A(n1n2...ni): i is an element of N} such that if M is simply bounded in one A(n1n2...ni) then M is uniformly bounded in A (Theorem 3.1). This result is deduced from the fact that if W = {E-n1n2...np: p, n(1), n(2),..., n(p) is an element of N} is a linear increasing web in I-0(x)(X, A), then there exists a strand {E-n1n2...ni: i is an element of N} such that every E-n1n2...ni is barrelled and dense in l(0)(x)(X, A) (Theorem 2.7). From this strong 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. (C) 1997 Academic Press.

[发布日期] 1997-06-01