On Grothendieck Sets

Abstract

We call a subset $\mathcal{M}$ of an algebra of sets $\mathcal{A}$ a Grothendieck set for the Banach space $ba\left(\mathcal{A}\right)$ of bounded finitely additive scalar-valued measures on $\mathcal{A}$ equipped with the variation norm if each sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(\mathcal{A}\right)$ which is pointwise convergent on $\mathcal{M}$ is weakly convergent in $ba\left(\mathcal{A}\right)$, i.e., if there is $\mu \in ba\left(\mathcal{A}\right)$ such that ${\mu }_n\left(A\right)\to \mu \left(A\right)$ for every $A\in \mathcal{M}$ then ${\mu }_n\to \mu$ weakly in $ba\left(\mathcal{A}\right)$. A subset $\mathcal{M}$ of an algebra of sets $\mathcal{A}$ is called a Nikodým set for $ba\left(\mathcal{A}\right)$ if each sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(\mathcal{A}\right)$ which is pointwise bounded on $\mathcal{M}$ is bounded in $ba\left(\mathcal{A}\right)$. We prove that if $\mathsf{\Sigma }$ is a $\sigma$-algebra of subsets of a set $\mathsf{\Omega }$ which is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets of $\mathsf{\Sigma }$ there exists $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is a Grothendieck set for $ba\left(\mathcal{A}\right)$. This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a $\sigma$-algebra $\mathsf{\Sigma }$ is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets, there is $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is a Nikodým set for $ba\left(\mathsf{\Sigma }\right)$. This also refines the Grothendieck result stating that for each $\sigma$-algebra $\mathsf{\Sigma }$ the Banach space ${\ell }_{\infty }\left(\mathsf{\Sigma }\right)$ is a Grothendieck space. Some applications to classic Banach space theory are given.

1. Introduction

With a different terminology, Valdivia showed in [1] that if a $\sigma$-algebra $\mathsf{\Sigma }$ of subsets of a set $\mathsf{\Omega }$ is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets, there is $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is a Nikodým set for $ba\left(\mathsf{\Sigma }\right)$. We prove that if $\mathsf{\Sigma }$ is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets of $\mathsf{\Sigma }$ there is $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is a Grothendieck set for $ba\left(\mathcal{A}\right)$ (definitions below). This statement is both the exact counterpart for Grothendieck sets of Valdivia’s result for Nikodým sets and a refinement of Grothendieck’s classic result stating that the Banach space ${\ell }_{\infty }\left(\mathsf{\Sigma }\right)$ of bounded scalar-valued $\mathsf{\Sigma }$-measurable functions defined on $\mathsf{\Omega }$ equipped with the supremum-norm is a Grothendieck space. Our previous result applies easily to Banach space theory to extend some well-known results. For example, Phillip’s lemma can be read as follows. If $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ is an increasing sequence of subsets of $\mathsf{\Sigma }$ covering $\mathsf{\Sigma }$, there is $p\in \mathbb{N}$ such that if ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }\subseteq ba\left(\mathsf{\Sigma }\right)$ verifies ${lim}_{n\to \infty }{\mu }_n\left(A\right)=0$ for every $A\in {\mathsf{\Sigma }}_p$ and $\left\{A_k:k\in \mathbb{N}\right\}$ is a sequence of pairwise disjoint elements of $\mathsf{\Sigma }$, then ${lim}_{n\to \infty }{\sum }_{k=1}^{\infty }\left|{\mu }_n\left(A_k\right)\right|=0$.

2. Preliminaries

In what follow we use the notation of [2] (Chapter 5). Let $\mathcal{R}$ be a ring of subsets of a nonempty set $\mathsf{\Omega }$, ${\chi }_A$ be the characteristic function of the set $A\in \mathcal{R}$ and let ${\ell }_0^{\infty }\left(\mathcal{R}\right)=\mathrm{span}\left\{{\chi }_A:A\in \mathcal{R}\right\}$ denote the linear space of all $\mathbb{K}$-valued $\mathcal{R}$-simple functions, $\mathbb{K}$ being the scalar field of real or complex numbers. Since $A\cap B\in \mathcal{R}$ and $A\Delta B\in \mathcal{R}$ whenever $A,B\in \mathcal{R}$, for each $f\in {\ell }_0^{\infty }\left(\mathcal{R}\right)$ there are pairwise disjoint sets $A_1,\dots ,A_m\in \mathcal{R}$ and nonzero $a_1,\dots ,a_m\in \mathbb{K}$, with $a_i\ne a_j$ if $i\ne j$ such that $f={\sum }_{i=1}^m{a}_i{\chi }_{A_i}$, with $f={\chi }_{\varnothing }$ if $f=0$. Unless otherwise stated we shall assume ${\ell }_0^{\infty }\left(\mathcal{R}\right)$ equipped with the norm ${∥f∥}_{\infty }=sup\left\{\left|f\left(\omega \right)\right|:\omega \in \mathsf{\Omega }\right\}$. If $Q=\mathrm{abx}\{{\chi }_A:A\in \mathcal{R}\}$ is the absolutely convex hull of $\{{\chi }_A:A\in \mathcal{R}\}$, an equivalent norm is defined on ${\ell }_0^{\infty }\left(\mathcal{R}\right)$ by the gauge of Q, namely ${∥f∥}_Q=inf\left\{\lambda >0:f\in \lambda Q\right\}$. For if $f\in {\ell }_0^{\infty }\left(\mathcal{R}\right)$ with ${∥f∥}_{\infty }\le 1$, it can be shown that $f\in 4Q$ (cf. [2] (Proposition 5.1.1)), hence ${∥·∥}_{\infty }\le {∥·∥}_Q\le 4{∥·∥}_{\infty }$.

The dual of ${\ell }_0^{\infty }\left(\mathcal{R}\right)$ is the Banach space $ba\left(\mathcal{R}\right)$ of bounded finitely additive scalar-valued measures on $\mathcal{R}$, which we shall assume to be equipped with the variation norm

$$\left|\mu \right|=sup\sum _{i=1}^n\left|\mu \left(E_i\right)\right|,$$

where the supremum is taken over all finite sequences of pairwise disjoint members of $\mathcal{R}$. This is the dual of the supremum-norm ${∥·∥}_{\infty }$ of ${\ell }_0^{\infty }\left(\mathcal{R}\right)$. An equivalent norm is given by $∥\mu ∥=sup\left\{\left|\mu \left(A\right)\right|:A\in \mathcal{R}\right\}$, which is the dual norm of the gauge ${∥·∥}_Q$. We shall also consider the Banach space $ba{\left(\mathcal{R}\right)}^{*}$ equipped with the bidual norm $∥·∥$ of ${∥·∥}_{\infty }$. The completion of the normed space $\left({\ell }_0^{\infty }\left(\mathcal{R}\right),{∥·∥}_{\infty }\right)$ is the Banach space ${\ell }_{\infty }\left(\mathcal{R}\right)$ of all bounded $\mathcal{R}$-measurable functions.

The Banach space ${\ell }_{\infty }\left(\mathcal{R}\right)$ embeds isometrically into $ba{\left(\mathcal{R}\right)}^{*}$, hence each characteristic function ${\chi }_A$ in ${\ell }_0^{\infty }\left(\mathcal{R}\right)$ with $A\in \mathcal{R}$ can be considered as a bounded linear functional on $ba\left(\mathcal{R}\right)$ defined by evaluation $⟨{\chi }_A,\mu ⟩=\mu \left(A\right)$. So, we may write $\left\{{\chi }_A:A\in \mathcal{R}\right\}\subseteq S_{ba{\left(\mathcal{R}\right)}^{*}}$, where $S_{ba{\left(\mathcal{R}\right)}^{*}}$ stands for the unit sphere of $ba{\left(\mathcal{R}\right)}^{*}$, and the set $\left\{{\chi }_A:A\in \mathcal{R}\right\}$, regarded as a topological subspace of $ba{\left(\mathcal{R}\right)}^{*}\left({\mathrm{weak}}^{*}\right)$, is the same as $\left\{{\chi }_A:A\in \mathcal{R}\right\}$ regarded as a topological subspace of ${\ell }_0^{\infty }\left(\mathcal{R}\right)\left(\mathrm{weak}\right)$.

A subfamily $ϝ$ of an algebra of sets $\mathcal{A}$ is called a Nikodým set for $ba\left(\mathcal{A}\right)$ (cf. [3]) if each set $\{{\mu }_{\alpha }:\alpha \in \mathsf{\Lambda }\}$ in $ba\left(\mathcal{A}\right)$ which is pointwise bounded on $ϝ$ is bounded in $ba\left(\mathcal{A}\right)$, i.e., if ${sup}_{\alpha \in \mathsf{\Lambda }}\left|{\mu }_{\alpha }\left(A\right)\right|<\infty$ for each $A\in ϝ$ implies that ${sup}_{\alpha \in \mathsf{\Lambda }}\left|{\mu }_{\alpha }\right|<\infty$. The algebra $\mathcal{A}$ is said to have property$\left(N\right)$ if the whole family $\mathcal{A}$ is a Nikodým set for $ba\left(\mathcal{A}\right)$. Nikodým’s classic boundedness theorem establishes that every $\sigma$-algebra has property $\left(N\right)$. An algebra $\mathcal{A}$ is said to have property $\left(G\right)$ if ${\ell }_{\infty }\left(\mathcal{A}\right)$ is a Grothendieck space, i.e., if each weak* convergent sequence in $ba\left(\mathcal{A}\right)$ is weakly convergent in the Banach space $ba\left(\mathcal{A}\right)$. The fact that every $\sigma$-algebra has property $\left(G\right)$ is also due to Grothendieck. Every countable algebra lacks property $\left(N\right)$, and the algebra $\mathfrak{J}$ of Jordan-measurable subsets of the real interval $[0,1]$ has property $\left(N\right)$ but fails property $\left(G\right)$ (cf. [4] (Propositions 3.2 and 3.3) and [5]). Let us recall that a sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(\mathcal{A}\right)$ is uniformly exhaustive if for each sequence $\left\{A_i:i\in \mathbb{N}\right\}$ of pairwise disjoint elements of $\mathcal{A}$ it holds that ${lim}_{k\to \infty }{sup}_{n\in \mathbb{N}}\left|{\mu }_n\left(A_k\right)\right|=0$. We shall use the following result, originally stated in [4] (2.3 Definition).

Theorem 1.

An algebra of sets $\mathcal{A}$ has property $\left(G\right)$ if and only if every bounded sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(\mathcal{A}\right)$ which converges pointwise on $\mathcal{A}$ is uniformly exhaustive.

An algebra $\mathcal{A}$ is said to have property $\left(VHS\right)$ if every sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(\mathcal{A}\right)$ which converges pointwise on $\mathcal{A}$ is uniformly exhaustive. It should be mentioned that $\left(VHS\right)\iff \left(N\right)\wedge \left(G\right)$, where the proof of the non-trivial implication can be found in [6] (see also [7] (Theorem 4.2)). For later use we introduce the following definition.

Definition 1.

A subfamily $\mathcal{M}$ of an algebra of sets $\mathcal{A}$ will be called a Grothendieck set for $ba\left(\mathcal{A}\right)$ if each sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(\mathcal{A}\right)$ which is pointwise convergent on $\mathcal{M}$ is weakly convergent in $ba\left(\mathcal{A}\right)$, i.e., if there is $\mu \in ba\left(\mathcal{A}\right)$ such that ${\mu }_n\left(A\right)\to \mu \left(A\right)$ for every $A\in \mathcal{M}$ then ${\mu }_n\to \mu$ weakly in $ba\left(\mathcal{A}\right)$.

If an algebra $\mathcal{A}$ contains a Grothendieck subset for $ba\left(\mathcal{A}\right)$, clearly $\mathcal{A}$ has property $\left(G\right)$. Grothendieck sets are closely related to the so-called Rainwater sets (defined below) for $ba\left(\mathcal{A}\right)$, and the study of the Rainwater sets for $ba\left(\mathcal{A}\right)$ leads to Theorem 4 below, from which the following result is a straightforward corollary.

Theorem 2.

If Σ is a σ-algebra of subsets of a set Ω which is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets of Σ there exists $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is a Grothendieck set for $ba\left(\mathsf{\Sigma }\right)$.

Indeed, in [1] (Theorem 1) Valdivia showed that if a $\sigma$-algebra $\mathsf{\Sigma }$ of subsets of a set $\mathsf{\Omega }$ is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets (subfamilies) of $\mathsf{\Sigma }$, there exists some $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is a Nikodým set for $ba\left(\mathsf{\Sigma }\right)$ or, equivalently, that given an increasing sequence $\left\{E_n:n\in \mathbb{N}\right\}$ of linear subspaces of ${\ell }_0^{\infty }\left(\mathsf{\Sigma }\right)$ covering ${\ell }_0^{\infty }\left(\mathsf{\Sigma }\right)$, there exists $p\in \mathbb{N}$ such that $E_p$ is dense and barrelled (see also [8] (Theorem 3)).

As a consequence of Theorem 4 we show that if a $\sigma$-algebra $\mathsf{\Sigma }$ is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets, there exists some $p\in \mathbb{N}$ such that $\left\{{\chi }_A:A\in {\mathsf{\Sigma }}_p\right\}$, regarded as a subset of the dual unit ball of $ba\left(\mathsf{\Sigma }\right)$, is also a Rainwater set for $ba\left(\mathsf{\Sigma }\right)$. This easily implies Theorem 2. In the last section we give some applications of Theorem 2 to classic Banach space theory which seems to have gone unnoticed so far. Let us point out that some results of this paper hold for Boolean algebras [9] (Theorem 12.35).

3. Rainwater Sets for $\mathit{b}\mathit{a}\left(\mathcal{A}\right)$

A subset X of the dual closed unit ball $B_{E^{*}}$ of a Banach space E is called a Rainwater set for E if every bounded sequence ${\left\{x_n\right\}}_{n=1}^{\infty }$ of E that converges pointwise on X, i.e., such that $x^{*}{x}_n\to x^{*}x$ for each $x^{*}\in X$, converges weakly in E (cf. [10]). Rainwater’s classic theorem [11] asserts that the set of the extreme points of the closed dual unit ball of a Banach space E is a Rainwater set for E. According to [12] (Corollary 11), each James boundary of E is a Rainwater set for E. As regards the Banach space $C\left(X\right)$ of real-valued continuous functions over a compact Hausdorff spaceX equipped with the supremum norm, if $K=\mathrm{Ext}{B}_{C{\left(X\right)}^{*}}$ is the set of the extreme points of the compact subset $B_{C{\left(X\right)}^{*}}$ of $C{\left(X\right)}^{*}\left({\mathrm{weak}}^{*}\right)$, the Arens-Kelly theorem asserts that $K=\left\{\pm {\delta }_x:x\in X\right\}$ (see [13]). By the Lebesgue dominated convergence theorem, if ${\left\{f_n\right\}}_{n=1}^{\infty }$ is a norm-bounded sequence in $C\left(X\right)$ (with respect to the supremum-norm) then $f_n\to f$ weakly in $C\left(X\right)$ if and only if $f_n\left(x\right)\to f\left(x\right)$ for every $x\in X$, that is, $⟨f_n,\mu ⟩\to ⟨f,\mu ⟩$ for every $\mu \in C{\left(X\right)}^{*}$ if and only if $⟨f_n,{\delta }_v⟩\to ⟨f,{\delta }_v⟩$ for each $v\in K$ (see [14] (IV.6.4 Corollary)). This is Rainwater’s theorem for $C\left(X\right)$. In [10] the weak K-analyticity of the Banach space $C^b\left(X\right)$ of real-valued continuous and bounded functions defined on a completely regular space X equipped with the supremum norm is characterized in terms of certain Rainwater sets for $C^b\left(X\right)$. The next theorem, based on [3] (Proposition 4.1), exhibits a connection between Rainwater sets and property $\left(G\right)$. We include it for future reference and provide a proof for the sake of completeness.

Theorem 3.

Let $\mathcal{A}$ be an algebra of sets. The following are equivalent

1. 

$\mathcal{A}$ has property $\left(G\right)$.

2. 

$\left\{{\chi }_A:A\in \mathcal{A}\right\}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$, considered as a subset of $ba{\left(\mathcal{A}\right)}^{*}$.

3. 

The unit ball of ${\ell }_0^{\infty }\left(\mathcal{A}\right)$ is a Rainwater set for $ba\left(\mathcal{A}\right)$.

4. 

The unit ball of ${\ell }_{\infty }\left(\mathcal{A}\right)$ is a Rainwater set for $ba\left(\mathcal{A}\right)$.

Proof. 

$1\Rightarrow 2$. Assume that $\mathcal{A}$ has property $\left(G\right)$ and let ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ be a bounded sequence in $ba\left(\mathcal{A}\right)$ and $\mu \in ba\left(\mathcal{A}\right)$ such that $⟨{\chi }_A,{\mu }_n⟩\to ⟨{\chi }_A,\mu ⟩$ for each $A\in \mathcal{A}$. i.e., such that ${\mu }_n\left(A\right)\to \mu \left(A\right)$ for each $A\in \mathcal{A}$. By Theorem 1 the sequence $M=\left\{{\mu }_n:n\in \mathbb{N}\right\}$ is (bounded and) uniformly exhaustive on $\mathcal{A}$, so [15] (Corollary 5.2) produces a nonnegative real-valued finitely-additive measure $\lambda$ on $\mathcal{A}$ such that ${lim}_{\lambda \left(E\right)\to 0}{sup}_{n\in \mathbb{N}}\left|{\mu }_n\left(E\right)\right|=0$. Hence, [14] (4.9.12 Theorem]) shows that M is relatively weakly sequentially compact. Given that ${\mu }_n\left(A\right)\to \mu \left(A\right)$ for each $A\in \mathcal{A}$, necessarily $\mu$ is the only possible weakly adherent point of the sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$. So we get that ${\mu }_n\to \mu$ weakly in $ba\left(\mathcal{A}\right)$, which shows that $\left\{{\chi }_A:A\in \mathcal{A}\right\}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$.

$2\Rightarrow 3$. If $B_{ba{\left(\mathcal{A}\right)}^{*}}$ denotes the second dual ball of the closed unit ball $B_{{\ell }_{\infty }\left(\mathcal{A}\right)}$ of ${\ell }_{\infty }\left(\mathcal{A}\right)$ and $B_0$ stands for the unit ball of ${\ell }_0^{\infty }\left(\mathcal{A}\right)$, from the relations $\left\{{\chi }_A:A\in \mathcal{A}\right\}\subseteq B_0\subseteq B_{ba{\left(\mathcal{A}\right)}^{*}}$ it follows that $B_0$ is a also Rainwater set for $ba\left(\mathcal{A}\right)$.

$3\Rightarrow 4$ is obvious.

$4\Rightarrow 1$. If ${\mu }_n\to \mu$ in $ba\left(\mathcal{A}\right)$ under the weak* topology $\sigma \left(ba\left(\mathcal{A}\right),{\ell }_{\infty }\left(\mathcal{A}\right)\right)$ of $ba\left(\mathcal{A}\right)$ then ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ is a bounded sequence in $ba\left(\mathcal{A}\right)$. Given that $⟨{\mu }_n,f⟩\to ⟨\mu ,f⟩$ for every $f\in B_{{\ell }_{\infty }\left(\mathcal{A}\right)}$ and given the hypothesis that $B_{{\ell }_{\infty }\left(\mathcal{A}\right)}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$, we have that ${\mu }_n\to \mu$ weakly in $ba\left(\mathcal{A}\right)$. Consequently $\mathcal{A}$ has property $\left(G\right)$. □

Example 1.

If$\mathcal{Z}$stands for the algebra generated by the sets of density zero in$\mathbb{N}$, then$\left\{{\chi }_A:A\in \mathcal{Z}\right\}$is not a Rainwater set for$ba\left(\mathcal{Z}\right)$. This follows from the previous theorem and from the fact that $\mathcal{Z}$does not have property $\left(G\right)$ (see [16]).

Theorem 4.

Assume that $\mathcal{A}$ is an algebra of sets. Let $\mathcal{M}$ be a Nikodým subset for $ba\left(\mathcal{A}\right)$ and let $\left\{{\mathcal{M}}_n:n\in \mathbb{N}\right\}$ be an increasing covering of $\mathcal{M}$ by subsets of $\mathcal{M}$. If $\left\{{\chi }_A:A\in \mathcal{M}\right\}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$, there exists some $p\in \mathbb{N}$ such that $\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$.

Proof. 

Assume that $\left\{{\chi }_A:A\in \mathcal{M}\right\}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$. First we claim that

$$\left\{{\chi }_A:A\in \mathcal{A}\right\}\subseteq \bigcup _{n=1}^{\infty }n·{\overline{\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_n\right\}}}^{{∥·∥}_{\infty }}$$

Let us proceed by contradiction. Assume otherwise that there exists $B\in \mathcal{A}$ such that ${\chi }_B\notin n·{\overline{\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_n\right\}}}^{{∥·∥}_{\infty }}$ for all $n\in \mathbb{N}$. In this case the separation theorem provides ${\mu }_n\in ba\left(\mathcal{A}\right)$ with $\left|{\mu }_n\left(B\right)\right|=1$ such that

$$sup\left\{\left|⟨f,{\mu }_n⟩\right|:f\in {\overline{\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_n\right\}}}^{{∥·∥}_{\infty }}\right\}\le \frac{1}{n}$$

So, in particular it holds that

$$sup\left\{\left|{\mu }_n\left(A\right)\right|:A\in {\mathcal{M}}_n\right\}\le \frac{1}{n}$$

for every $n\in \mathbb{N}$. If $M\in \mathcal{M}$ there is $k\in \mathbb{N}$ such that $M\subseteq {\mathcal{M}}_n$ for every $n\ge k$. Consequently $\left|{\mu }_n\left(M\right)\right|\le \frac{1}{n}$ for $n\ge k$, which shows that ${\mu }_n\left(M\right)\to 0$. Since $\mathcal{M}$ is a Nikodým set and ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ is pointwise bounded on $\mathcal{M}$, it follows that ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ is bounded in $ba\left(\mathcal{A}\right)$. So, the fact that ${\mu }_n\left(M\right)\to 0$ for all $M\in \mathcal{M}$ along with the assumption that $\mathcal{M}$ is a Rainwater set leads to ${\mu }_n\to 0$ weakly in $ba\left(\mathcal{A}\right)$. This is a contradiction, since $⟨{\chi }_B,{\mu }_n⟩={\mu }_n\left(B\right)=1$ for every $n\in \mathbb{N}$. The claim is proved.

Set $Q:=\left\{{\chi }_A:A\in \mathcal{A}\right\}$. Since we are assuming that $\mathcal{M}$ is a Nikodým set for $ba\left(\mathcal{A}\right)$, the larger set $\mathcal{A}$ is also a Nikodým set for $ba\left(\mathcal{A}\right)$, which implies that ${\ell }_0^{\infty }\left(\mathcal{A}\right)$ is a metrizable barrelled space, hence a Baire-like space (see [17]). On the other hand, as a consequence of the previous claim, the family ${\left\{W_n\right\}}_{n=1}^{\infty }$ with

$$W_n:=n·{\overline{\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_n\right\}}}^{{∥·∥}_{\infty }}$$

is an increasing sequence of closed absolutely convex sets covering ${\ell }_0^{\infty }\left(\mathcal{A}\right)$. So, there exists $p\in \mathbb{N}$ such that

$$Q\subseteq p·{\overline{\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}}}^{{∥·∥}_{\infty }},$$

which shows that

$${\overline{\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}}}^{{∥·∥}_{\infty }}$$

is a Rainwater set for $ba\left(\mathcal{A}\right)$.

We claim that this implies that $\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$. In order to establish the claim it suffices to show that $\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$. So, let ${\left\{{\lambda }_n\right\}}_{n=1}^{\infty }$ be a bounded sequence in $ba\left(\mathcal{A}\right)$ such that $⟨u,{\lambda }_n⟩\to 0$ for every $u\in \mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}$. Let us show that $⟨v,{\lambda }_n⟩\to 0$ for each $v\in {\overline{\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}}}^{{∥·∥}_{\infty }}$. If $v\in {\overline{\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}}}^{{∥·∥}_{\infty }}$ there exists a sequence ${\left\{u_k\right\}}_{k=1}^{\infty }$ in $\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}$ such that ${∥u_k-v∥}_{\infty }\to 0$. Consequently, given $ϵ>0$ there is $k\left(ϵ\right)\in \mathbb{N}$ with

$${∥u_{k\left(ϵ\right)}-v∥}_{\infty }<\frac{ϵ}{2\left(1+{sup}_{n\in \mathbb{N}}\left|{\lambda }_n\right|\right)}.$$

Let $n\left(ϵ\right)\in \mathbb{N}$ be such that

$$\left|⟨u_{k\left(ϵ\right)},{\lambda }_n⟩\right|<\frac{ϵ}{2}$$

for every $n\ge n\left(ϵ\right)$. Consequently, one has

$$\left|⟨v,{\lambda }_n⟩\right|\le \left|⟨v-u_{k\left(ϵ\right)},{\lambda }_n⟩\right|+\left|⟨u_{k\left(ϵ\right)},{\lambda }_n⟩\right|\le {∥u_{k\left(ϵ\right)}-v∥}_{\infty }\left|{\lambda }_n\right|+\left|⟨u_{k\left(ϵ\right)},{\lambda }_n⟩\right|<ϵ$$

for all $n\ge n_0\left(ϵ\right)$. This proves that $⟨v,{\lambda }_n⟩\to 0$ for each $v\in {\overline{\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}}}^{{∥·∥}_{\infty }}$. Since we have shown before that ${\overline{\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}}}^{{∥·∥}_{\infty }}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$, we get that ${\lambda }_n\to 0$ weakly in $ba\left(\mathcal{A}\right)$. Therefore the absolutely convex set $\mathrm{abx}\left\{{\chi }_A:A\in {\mathcal{M}}_p\right\}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$, a stated. □

Corollary 1.

Let $\mathcal{A}$ be an algebra of sets with property $\left(VHS\right)$. If $\left\{{\mathcal{A}}_n:n\in \mathbb{N}\right\}$ is an increasing covering of $\mathcal{A}$ consisting of subsets of $\mathcal{A}$, there is some $p\in \mathbb{N}$ such that $\left\{{\chi }_A:A\in {\mathcal{A}}_p\right\}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$.

Proof. 

This is a straightforward consequence of the Theorem 4 for $\mathcal{M}=\mathcal{A}$, since as mentioned earlier an algebra $\mathcal{A}$ has property $\left(VHS\right)$ if and only if $\mathcal{A}$ has both properties $\left(N\right)$ and $\left(G\right)$ (this also can be found in [7] (Theorem 4.2)). So, on the one hand $\mathcal{A}$ is a Nikodým set for $ba\left(\mathcal{A}\right)$ and, on the other hand, according to Theorem 3, the family $\left\{{\chi }_A:A\in \mathcal{A}\right\}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$. □

Proof of Theorem 2.

If $\mathsf{\Sigma }$ is a $\sigma$-algebra of subsets of a set $\mathsf{\Omega }$ which is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets of $\mathsf{\Sigma }$, Corollary 1 and Valdivia’s result [1] provide an index $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is a Nikodým set for $ba\left(\mathsf{\Sigma }\right)$ at the same time that $\left\{{\chi }_A:A\in {\mathsf{\Sigma }}_p\right\}$ is a Rainwater set for $ba\left(\mathsf{\Sigma }\right)$. If ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ verifies that ${\mu }_n\left(A\right)\to \mu \left(A\right)$ for every $A\in {\mathsf{\Sigma }}_p$, the sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ is bounded in $ba\left(\mathsf{\Sigma }\right)$ since ${\mathsf{\Sigma }}_p$ is a Nikodým set for $ba\left(\mathsf{\Sigma }\right)$. But then ${\mu }_n\to \mu$ weakly in $ba\left(\mathsf{\Sigma }\right)$ due to $\left\{{\chi }_A:A\in {\mathsf{\Sigma }}_p\right\}$ is a Rainwater set for $ba\left(\mathsf{\Sigma }\right)$. Consequently ${\mathsf{\Sigma }}_p$ is a Grothendieck for $ba\left(\mathsf{\Sigma }\right)$ and we are done. □

Corollary 2.

If $\left\{{\mathsf{\Lambda }}_n:n\in \mathbb{N}\right\}$ is an increasing sequence of subsets of $\mathsf{\Sigma }=2^{\mathbb{N}}$ covering $2^{\mathbb{N}}$, there exists some $p\in \mathbb{N}$ such that each sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(2^{\mathbb{N}}\right)$ that converges pointwise on ${\mathsf{\Lambda }}_p$ converges weakly in $ba\left(2^{\mathbb{N}}\right)={\ell }_{\infty }^{*}$.

Proof. 

Apply Theorem 2 to the $\sigma$-algebra $2^{\mathbb{N}}$. □

We complete our study of Rainwater sets for $ba\left(\mathcal{A}\right)$ with the following result. Note that if ${\overline{X}}^{w^{*}}$ (weak* closure) with $X\subseteq B_{ba{\left(\mathcal{A}\right)}^{*}}$ is a Rainwater set for $ba\left(\mathcal{A}\right)$ then X could not be a Rainwater set for $ba\left(\mathcal{A}\right)$. However the following property holds.

Theorem 5.

Let $\mathcal{A}$ be an algebra of sets. Assume that $\left\{{\chi }_A:A\in \mathcal{A}\right\}$ is a Grothendieck set for $ba\left(\mathcal{A}\right)$. If $\left\{{\chi }_A:A\in \mathcal{M}\right\}$ is a $G_{\delta }$-dense subset of $\left\{{\chi }_A:A\in \mathcal{A}\right\}$ under the relative weak* topology of $ba{\left(\mathcal{A}\right)}^{*}$ or, which is the same, under the relative weak topology of ${\ell }_0^{\infty }\left(\mathcal{A}\right)$, then $\left\{{\chi }_A:A\in \mathcal{M}\right\}$ is a Grothendieck set for $ba\left(\mathcal{A}\right)$.

Proof. 

Let ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ be a sequence in $ba\left(\mathcal{A}\right)$ such that ${\mu }_n\left(Q\right)\to 0$ for every $Q\in \mathcal{M}$. Given $B\in \mathcal{A}$, let us define $G_n:=\left\{{\chi }_C:C\in \mathcal{A},{\mu }_n\left(C\right)={\mu }_n\left(B\right)\right\}$. Then one has that ${\chi }_B\in {\bigcap }_{n=1}^{\infty }{G}_n$, so that $G:={\bigcap }_{n=1}^{\infty }{G}_n$ is a nonempty intersection of countably many zero-sets of $\left\{{\chi }_A:A\in \mathcal{A}\right\}$, hence a non-empty $G_{\delta }$-set in $\left\{{\chi }_A:A\in \mathcal{A}\right\}$ in the relative weak topology of ${\ell }_0^{\infty }\left(\mathcal{A}\right)$. According to the hypothesis G meets $\left\{{\chi }_A:A\in \mathcal{M}\right\}$. Hence there exists $M_B\in \mathcal{M}$ such that ${\chi }_{M_B}\in G\cap \left\{{\chi }_A:A\in \mathcal{M}\right\}$, which means that ${\mu }_n\left(M_B\right)={\mu }_n\left(B\right)$ for every $n\in \mathbb{N}$. Since ${\mu }_n\left(M_B\right)\to 0$, it follows that ${\mu }_n\left(B\right)\to 0$. So, we conclude that ${\mu }_n\left(B\right)\to 0$ for every $B\in \mathcal{A}$. Putting together that $\left(i\right)$$\left\{{\chi }_A:A\in \mathcal{A}\right\}$ is a Grothendieck set for $ba\left(\mathcal{A}\right)$, and $\left(ii\right)$ ${\mu }_n\left(B\right)\to 0$ for all $B\in \mathcal{A}$, we get that ${\mu }_n\to 0$ weakly in $ba\left(\mathcal{A}\right)$. Thus $\left\{{\chi }_A:A\in \mathcal{M}\right\}$ is a Grothendieck set for $ba\left(\mathcal{A}\right)$. □

4. Application to Banach Spaces

Theorem 2 facilitates the extension of various classic theorems of Banach space theory. As a sample, we include three of them: namely, the Phillips lemma about convergence in $ba\left(\mathsf{\Sigma }\right)$, Nikodým’s pointwise convergence theorem in $ca\left(\mathsf{\Sigma }\right)$ and the usual characterization of weak convergence in $ca\left(\mathsf{\Sigma }\right)$, the linear subspace of $ba\left(\mathsf{\Sigma }\right)$ consisting of the countably additive measures in $\mathsf{\Sigma }$ (see [18] (Chapter 7)).

Proposition 1.

Let Σ be a σ-algebra of subsets of a set Ω. If $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ is an increasing sequence of subsets of Σ covering Σ, there exists some $p\in \mathbb{N}$ enjoying the following property. If ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }\subseteq ba\left(\mathsf{\Sigma }\right)$ verifies ${lim}_{n\to \infty }{\mu }_n\left(A\right)=0$ for every $A\in {\mathsf{\Sigma }}_p$ and $\left\{A_k:k\in \mathbb{N}\right\}$ is a sequence of pairwise disjoint elements of Σ, then

$$\underset{n\to \infty }{lim}\sum _{k=1}^{\infty }\left|{\mu }_n\left(A_k\right)\right|=0.$$

(1)

Proof. 

According to Theorem 2 there is $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is Grothendieck set for $ba\left(\mathsf{\Sigma }\right)$. So, if ${lim}_{n\to \infty }{\mu }_n\left(A\right)=0$ for every $A\in {\mathsf{\Sigma }}_p$, then ${\mu }_n\to 0$ weakly in $ba\left(\mathsf{\Sigma }\right)$. In particular, ${\mu }_n\left(A\right)\to 0$ for every $A\in \mathsf{\Sigma }$. Hence, (1) holds by Phillip’s classic theorem. □

Proposition 2.

Let Σ be a σ-algebra of subsets of a set Ω. If $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ is an increasing sequence of subsets of Σ covering Σ, there exists some $p\in \mathbb{N}$ such that if ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }\subseteq ca\left(\mathsf{\Sigma }\right)$ verifies that ${\mu }_n\left(A\right)\to \mu \left(A\right)$ for every $A\in {\mathsf{\Sigma }}_p$ then the set $\left\{{\mu }_n:n\in \mathbb{N}\right\}$ is uniformly exhaustive and $\mu \in ca\left(\mathsf{\Sigma }\right)$.

Proposition 3.

Let Σ be a σ-algebra of subsets of a set Ω. If $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ is an increasing sequence of subsets of Σ covering Σ, there exists some $p\in \mathbb{N}$ such that ${\mu }_n\to \mu$ weakly in $ca\left(\mathsf{\Sigma }\right)$ if and only if ${\mu }_n\left(A\right)\to \mu \left(A\right)$ for every $A\in {\mathsf{\Sigma }}_p$.