文摘
Following Schachermayer, a subset B of an algebra e4c3b7e8009b7a7" title="Click to view the MathML source">A of subsets of Ω is said to have the N-property if a B-pointwise bounded subset M of ba(A) is uniformly bounded on e4c3b7e8009b7a7" title="Click to view the MathML source">A, where ba(A) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on e4c3b7e8009b7a7" title="Click to view the MathML source">A. Moreover B is said to have the strong N-property if for each increasing countable covering (Bm)m of B there exists e6" title="Click to view the MathML source">Bn which has the N-property. The classical Nikodym–Grothendieck's theorem says that each σ -algebra S of subsets of Ω has the N-property. The Valdivia's theorem stating that each σ -algebra S has the strong N -property motivated the main measure-theoretic result of this paper: We show that if 8bf8d9c6c8bad3f35" title="Click to view the MathML source">(Bm1)m1 is an increasing countable covering of a σ -algebra S and if 9f98cd89ef429ae10eb94f61a5" title="Click to view the MathML source">(Bm1,m2,…,mp,mp+1)mp+1 is an increasing countable covering of 9f572e21c7c02ee83f96710334cc78ec" title="Click to view the MathML source">Bm1,m2,…,mp, for each 9aad112a2eec5eab4d" title="Click to view the MathML source">p,mi∈N, e64eb8b29ac47d586e6444ea38e3ed" title="Click to view the MathML source">1⩽i⩽p, then there exists a sequence 9f92bc33eb5fa3643d" title="Click to view the MathML source">(ni)i such that each e67a0f" title="Click to view the MathML source">Bn1,n2,…,nr, r∈N, has the strong N -property. In particular, for each increasing countable covering (Bm)m of a σ -algebra S there exists e6" title="Click to view the MathML source">Bn which has the strong N-property, improving mentioned Valdivia's theorem. Some applications to localization of bounded additive vector measures are provided.