A Survey on Valdivia Open Question on Nikodým Sets

Let A be an algebra of subsets of a set Ω and b a ( 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 b a ( A ) if each countable B -pointwise bounded subset M of b a ( A ) is norm bounded. A subset B of A is a Grothendieck set for b a ( A ) if for each bounded sequence μ n n = 1 ∞ in b a ( A ) the B -pointwise convergence on b a ( A ) implies its b a ( A ) * -pointwise convergence on b a ( A ) . A subset B of an algebra A is a strong-Nikodým (Grothendieck) set for b a ( A ) if in each increasing covering { B n : n ∈ N } of B there exists B m which is a Nikodým (Grothendieck) set for b a ( 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 b a ( A ) then A is a strong Nikodým set for b a ( 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.