LAUSR

Abstract Following Schachermayer, a subset B of an algebra A of subsets of Ω is said to have the N-property if a B -pointwise bounded subset M of b a ( A ) is uniformly bounded on A , where b a ( A ) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on A . Moreover B is said to have the strong N-property if for each increasing countable covering ( B m ) m of B there exists B n 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 ( B m 1 ) m 1 is an increasing countable covering of a σ-algebra S and if ( B m 1 , m 2 , … , m p , m p + 1 ) m p + 1 is an increasing countable covering of B m 1 , m 2 , … , m p , for each p , m i ∈ N , 1 ⩽ i ⩽ p , then there exists a sequence ( n i ) i such that each B n 1 , n 2 , … , n r , r ∈ N , has the strong N-property. In particular, for each increasing countable covering ( B m ) m of a σ-algebra S there exists B n which has the strong N-property, improving mentioned Valdivia's theorem. Some applications to localization of bounded additive vector measures are provided.