Abstract
Let $\mathcal{A}$ be an algebra of subsets of a set $\Omega $, let $ba(\mathcal{A})$ the Banach space of real or complex bounded finitely additive measures defined on $\mathcal{A}$ endowed with the variation norm, which is equivalent to supremum norm, and let $\mathcal{B}$ a subset of $\mathcal{A}$. The subset $\mathcal{B}$ is a Nikodym set for\textit{\}$ba(\mathcal{A})$, or $\mathcal{B}$ has property $(N)$, if each $\mathcal{B}$-poinwise bounded sequence of the Banach space $ba(\mathcal{A})$ is bounded in $ba(\mathcal{A}) $, i.e., $\mathcal{B}$-poinwise boundedness imply uniform boundedness. $\mathcal{B}$ has property $(G)$ [$(VHS)$] if for each bounded sequence [if for each sequence] in $ba(\mathcal{A})$ the $\mathcal{B}$-poinwise convergence implies its weak convergence.
$\mathcal{B}$ has property $(sN)$ [$(sG)$ or $(sVHS)$] if for every
increasing covering $\{\mathcal{B}_{n}:n\in\mathbb{N}\}$ of $\mathcal{B}$ there exist $p\in\mathbb{N}$ such that $\mathcal{B}_{p}$ has property $(N)$ [$(G)$ or $(VHS)$], respectively, and, finally, $\mathcal{B}$ has property $(wN)$ [$(wG)$ or $(wVHS)$] if every increasing web $\{\mathcal{B}_{n_{1}n_{2}\cdots n_{m}}:n_{i}\in\mathbb{N},1\leq i\leq m,m\in\mathbb{N}\}$ of $\mathcal{B}$ there exists a sequence $\left(p_{n}:n\in\mathbb{N}\right)$ such that each $\mathcal{B}_{p_{1}p_{2}\cdots p_{m}}$ has property
$(N)$ [$(G)$ or $(VHS)$], respectively, for every $m\in\mathbb{N}$.
The classical theorems of Nikodym-Grothendieck, Valdivia, Grothendieck and Vitali-Hahn-Saks say that every $\sigma $-algebra $\mathcal{A}$ of subsets of a set $\Omega $ has the properties $(N)$, $(sN)$, $(G)$ and $(VHS)$.
The aim of this brief survey is to gather the proof that Every $\sigma$-algebra of subsets of a set $\Omega $ has property $(wN)$ with some simplifications and to show the role of this property to prove that a subset $\mathcal{B}$ of an algebra $\mathcal{A}$ has property $(wWHS)$ if and only if $\mathcal{B}$ has property $(wN)$ and $\mathcal{A}$ has property $(G)$. Some open questions will be considered.
