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A subset B of an algebra A of subsets of a set Ω has property (N) if each B-pointwise bounded sequence of the Banach space ba(A) is bounded in ba(A), where ba(A) is the Banach space of real or complex bounded finitely additive measures defined on A endowed with the variation norm. B has property (G) [(VHS)] if for each bounded sequence [if for each sequence] in ba(A) the B-pointwise convergence implies its weak convergence. B has property (sN) [(sG) or (sVHS)] if every increasing covering {Bn:n∈N} of B contains a set Bp with property (N) [(G) or (VHS)], and B has property (wN) [(wG) or (wVHS)] if every increasing web {Bn1n2⋯nm:ni∈N,1≤i≤m,m∈N} of B contains a strand {Bp1p2⋯pm:m∈N} formed by elements Bp1p2⋯pm with property (N) [(G) or (VHS)] for every m∈N. The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every σ-algebra has properties (N), (sN), (G) and (VHS). Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every σ-algebra has property (wN) and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property (wN) of a σ-algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset B of an algebra A has property (wWHS) if and only if B has property (wN) and A has property (G).

[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., Research supported for the second named author by Grant PGC2018-094431-B-I00 of Ministry of Science, Innovation & Universities of Spain.

A subset $$\mathscr {B}$$ of an algebra $$\mathscr {A}$$ of subsets of $$\varOmega $$ is a Nikodým set for $$ba(\mathscr {A})$$ if each $$\mathscr {B}$$ -pointwise bounded subset M of $$ba(\mathscr {A})$$ is uniformly bounded on $$\mathscr {A}$$ and $$\mathscr {B}$$ is a strong Nikodým set for $$ba(\mathscr {A})$$ if each increasing covering $$(\mathscr {B}_{m})_{m=1}^{\infty }$$ of $$\mathscr {B}$$ contains a $$\mathscr {B}_{n}$$ which is a Nikodým set for $$ba(\mathscr {A})$$ , where $$ba(\mathscr {A})$$ is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on $$\mathscr {A}$$ . The subset $$\mathscr {B}$$ has (VHS) property if $$\mathscr {B}$$ is a Nikodým set for $$ba(\mathscr {A})$$ and for each sequence $$(\mu _{n})_{m=1}^{\infty }$$ and each $$\mu $$ , both in $$ba(\mathscr {A})$$ and such that $$\lim _{n\rightarrow \infty }\mu _{n}(B)=\mu (B)$$ , for each $$B\in \mathscr {B}$$ , we have that the sequence $$(\mu _{n})_{m=1}^{\infty }$$ converges weakly to $$\mu $$ . We prove that if $$(\mathscr {B} _{m})_{m=1}^{\infty }$$ is an increasing covering of and algebra $$\mathscr {A}$$ that has (VHS) property and there exist a $$\mathscr {B}_{n}$$ which is a Nikodým set for $$ba(\mathscr {A})$$ then there exists $$\mathscr {B}_{q}$$ , with $$q\ge p$$ , such that $$\mathscr {B}_{q}$$ has (VHS) property. In particular, if $$(\mathscr {B}_{m})_{m=1}^{\infty }$$ is an increasing covering of a $$\sigma $$ -algebra there exists $$\mathscr {B}_{q}$$ that has (VHS) property. Valdivia proved that every $$\sigma $$ -algebra has strong Nikodým property and in 2013 asked if Nikodým property in an algebra implies strong Nikodým property. We present three open questions related with this aforementioned Valdivia question and a proof of his strong Nikodým Theorem for $$\sigma $$ -algebras that it is independent of the Barrelled spaces theory and it is developed with basic results of Measure theory and Banach spaces.