On Schachermayer and Valdivia results in algebras of Jordan measurable sets

Abstract

[EN] A subset B of a set-algebra A has property N if each B-pointwise bounded subset M of bounded measures of bounded variation is A -uniformly bounded. Each &#963;-algebra has property N and there exists algebras which does not have property N, for instance, the algebra of finite and co-finite subsets of N. Schachermayer proved that the algebra J(I ) of Jordan measurable subsets of I := [0, 1] has property N and J(I ) is not a &#963;-algebra. In 2013, Valdivia proved that if (B_m_1 )_m_1 is an increasing countable covering of the algebra J(K) of Jordan measurable subsets of a product K of k icompact intervals [a_i , b_i], 1<=i<=k, there exists B_n_1 which has property N. In this paper we prove that if, additionally, for each m_1 &#8712; N, (m_1,m_2) &#8712; N^2, the sequences (B_m_1,p_2 )_p_2, (B_m_1,m_2,p_3)_p_3, are increasing coverings of B_m_1 , B_m_1,m_2, , respectively, there exists a sequence (n_r)_r such that each B_n_1,n_2;...;n_r has property N. This result extends the mentioned Schachermayer and Valdivia results and enables us to give some applications to bounded vector measures.