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.

中文翻译：

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集合代数覆盖中的 Vitali-Hahn-Saks 性质

代数 $$\mathscr {A}$$ 的 $$\varOmega $$ 子集的子集 $$\mathscr {B}$$ 是 $$ba(\mathscr {A})$$ 的 Nikodým 集，如果$$ba(\mathscr {A})$$ 的每个 $$\mathscr {B}$$ -pointwise 有界子集 M 在 $$\mathscr {A}$$ 和 $$\mathscr {B}$ 上一致有界$ 是 $$ba(\mathscr {A})$$ 的强 Nikodým 集，如果每个递增覆盖 $$(\mathscr {B}_{m})_{m=1}^{\infty }$$ $$\mathscr {B}$$ 包含 $$\mathscr {B}_{n}$$，它是 $$ba(\mathscr {A})$$ 的 Nikodým 集，其中 $$ba(\mathscr {A})$$ 是 $$\mathscr {A}$$ 上定义的有界变化的实（或复）有限加法测度的 Banach 空间。如果 $$\mathscr {B}$$ 是为 $$ba(\mathscr {A})$$ 和每个序列 $$( \mu _{n})_{m=1}^{\infty }$$ 和每个 $$\mu $$ ，在 $$ba(\mathscr {A})$$ 和 $$\lim _{n\rightarrow \infty }\mu _{n}(B)=\mu (B)$$ 中，对于每个 $ $B\in \mathscr {B}$$ ，我们有序列 $$(\mu _{n})_{m=1}^{\infty }$$ 弱收敛到 $$\mu $$ 。我们证明如果 $$(\mathscr {B} _{m})_{m=1}^{\infty }$$ 是代数 $$\mathscr {A}$$ 的递增覆盖) 属性，并且存在 $$\mathscr {B}_{n}$$ 是 Nikodým 集 $$ba(\mathscr {A})$$ 那么存在 $$\mathscr {B}_{q }$$ ，与 $$q\ge p$$ ，这样 $$\mathscr {B}_{q}$$ 具有（VHS）属性。特别地，如果 $$(\mathscr {B}_{m})_{m=1}^{\infty }$$ 是 $$\sigma $$ -代数的递增覆盖，则存在 $$\mathscr {B}_{q}$$ 具有 (VHS) 属性。Valdivia 证明了每个 $$\sigma $$ -代数都具有强 Nikodým 性质，并在 2013 年询问代数中的 Nikodým 性质是否意味着强 Nikodým 性质。我们提出了三个与上述 Valdivia 问题相关的开放性问题，以及他对 $$\sigma $$ -algebras 的强 Nikodým 定理的证明，即它独立于桶空间理论，并且它是根据测度理论和巴拿赫空间的基本结果开发.