A Survey on Valdivia Open Question on Nikodým Sets

Abstract

Let A be an algebra of subsets of a set Ω and ba(A) the Banach space of bounded finitely additive scalar-valued measures on A endowed with the variation norm. A subset B of A is a Nikodým set for ba(A) if each countable B-pointwise bounded subset M of ba(A) is norm bounded. A subset B of A is a Grothendieck set for ba(A) if for each bounded sequence μnn=1∞ in ba(A) the B-pointwise convergence on ba(A) implies its ba(A)*-pointwise convergence on ba(A). A subset B of an algebra A is a strong-Nikodým (Grothendieck) set for ba(A) if in each increasing covering {Bn:n∈N} of B there exists Bm which is a Nikodým (Grothendieck) set for ba(A). The answer of the following open question for an algebra A of subsets of a set Ω, proposed by Valdivia in 2013, has not yet been found: Is it true that if A is a Nikodým set for ba(A) then A is a strong Nikodým set for ba(A)? In this paper we surveyed some results related to this Valdivia’s open question, as well as the corresponding problem for strong Grothendieck sets. The new Propositions 1 and 3 provide more simplified proofs, particularly in their application to Theorems 1 and 2, which were the main results surveyed. Moreover, the proofs of almost all other propositions are wholly or partially original.