文摘
Following Schachermayer, a subset 8453b9e73957c3" title="Click to view the MathML source">B of an algebra A of subsets of Ω is said to have the N-property if a 8453b9e73957c3" title="Click to view the MathML source">B-pointwise bounded subset M of bb3380556" title="Click to view the MathML source">ba(A) is uniformly bounded on A, where bb3380556" title="Click to view the MathML source">ba(A) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on A. Moreover 8453b9e73957c3" title="Click to view the MathML source">B is said to have the strong N-property if for each increasing countable covering 9c5eb1194df64ed44bffab98ede" title="Click to view the MathML source">(Bm)m of 8453b9e73957c3" title="Click to view the MathML source">B there exists Bn which has the N-property. The classical Nikodym–Grothendieck's theorem says that each σ -algebra 9c9cb4f6294129d5" title="Click to view the MathML source">S of subsets of Ω has the N-property. The Valdivia's theorem stating that each σ -algebra 9c9cb4f6294129d5" title="Click to view the MathML source">S has the strong N -property motivated the main measure-theoretic result of this paper: We show that if 9c6c8bad3f35" title="Click to view the MathML source">(Bm1)m1 is an increasing countable covering of a σ -algebra 9c9cb4f6294129d5" title="Click to view the MathML source">S and if 9ef429ae10eb94f61a5" title="Click to view the MathML source">(Bm1,m2,…,mp,mp+1)mp+1 is an increasing countable covering of Bm1,m2,…,mp, for each p,mi∈N, 1⩽i⩽p, then there exists a sequence (ni)i such that each Bn1,n2,…,nr, r∈N, has the strong N -property. In particular, for each increasing countable covering 9c5eb1194df64ed44bffab98ede" title="Click to view the MathML source">(Bm)m of a σ -algebra 9c9cb4f6294129d5" title="Click to view the MathML source">S there exists Bn which has the strong N-property, improving mentioned Valdivia's theorem. Some applications to localization of bounded additive vector measures are provided.