A Survey on Nikodým and Vitali-Hahn-Saks Properties

Abstract

[EN] Let ba(A) be the Banach space of the real (or complex) finitely additive measures of bounded variation defined on an algebra A of subsets of Omega and endowed with the variation norm. . A subset B of A is a Nikod ́ym set for ba(A) if each B-pointwise bounded subset M of ba(A) is uniformly bounded on A and B is a strong Nikod´ym set for ba(A) if each increasing covering (Bm)1 m=1 of B contains a Bn which is a Nikod´ym set for ba(A). If, additionally, the Nikod´ym subset B verifies that the sequential B-pointwise convergence in ba(A) implies weak convergence then B has the Vitali-Hahn-Saks property, (VHS ) in brief, and B has the strong (VHS ) property if for each increasing covering (Bm)1m=1 of B there exists Bq that has (VHS ) property Motivated by Valdivia result that every -algebra has strong Nikod´ym property and by his 2013 open question concerning that if Nikod´ym property in an algebra of subsets implies strong Nikod´ym property we survey this Valdivia theorem and we get that in a strong Nikod´ym set the (VHS ) property implies the strong (VHS ) property.