Nikodym boundedness property for webs in sigma-algebras

Abstract

[EN] A subset B of an algebra A of subsets of Omega is said to have property N if a B-pointwise bounded subset M of ba(A ) is uniformly bounded on A , where ba(A ) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on A with the norm variation. Moreover B is said to have property sN if for each increasing countable covering (B_m)_m of B there exists B_n which has property N and B is said to have property wN if given the increasing countable coverings (B_m_1 )_m_1 of B and (B_m_1m_2...m_pm_(p+1) )_m_(p+1) of B_m_1m_2...m_p , for each p,m_i &#8712; N, 1<= i <= p + 1, there exists a sequence (n_i )_i such that each B_n_1n_2...n_r , r &#8712; N, has property N. For a &#963;-algebra S of subsets of Omega it has been proved that S has property N (Nikodym Grothendieck), property sN (Valdivia) and property w(sN) (Kakol López-Pellicer). We give a proof of property wN for a &#963;-algebra S which is independent of properties N and sN. This result and the equivalence of properties wN and w2N enable us to give some applications to localization of bounded additive vector measures.