On Valdivia strong version of Nikodym boundedness property

Following Schachermayer, a subset BB of an algebra AA of subsets of Ω is said to have the N-property   if a BB-pointwise bounded subset M   of ba(A)ba(A) is uniformly bounded on AA, where ba(A)ba(A) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on AA. Moreover BB is said to have the strong N-property   if for each increasing countable covering (Bm)m(Bm)m of BB there exists BnBn which has the N-property. The classical Nikodym–Grothendieck's theorem says that each σ  -algebra SS of subsets of Ω has the N-property. The Valdivia's theorem stating that each σ  -algebra SS has the strong N  -property motivated the main measure-theoretic result of this paper: We show that if (Bm1)m1(Bm1)m1 is an increasing countable covering of a σ  -algebra SS and if (Bm1,m2,…,mp,mp+1)mp+1(Bm1,m2,…,mp,mp+1)mp+1 is an increasing countable covering of Bm1,m2,…,mpBm1,m2,…,mp, for each p,mi∈Np,mi∈N, 1⩽i⩽p1⩽i⩽p, then there exists a sequence (ni)i(ni)i such that each Bn1,n2,…,nrBn1,n2,…,nr, r∈Nr∈N, has the strong N  -property. In particular, for each increasing countable covering (Bm)m(Bm)m of a σ  -algebra SS there exists BnBn which has the strong N-property, improving mentioned Valdivia's theorem. Some applications to localization of bounded additive vector measures are provided.