On Grothendieck Sets

: We call a subset M of an algebra of sets A a Grothendieck set for the Banach space ba ( A ) of bounded ﬁnitely additive scalar-valued measures on A equipped with the variation norm if each sequence { µ n } ∞ n = 1 in ba ( A ) which is pointwise convergent on M is weakly convergent in ba ( A ) , i. e., if there is µ ∈ ba ( A ) such that µ n ( A ) → µ ( A ) for every A ∈ M then µ n → µ weakly in ba ( A ) . A subset M of an algebra of sets A is called a Nikodým set for ba ( A ) if each sequence { µ n } ∞ n = 1 in ba ( A ) which is pointwise bounded on M is bounded in ba ( A ) . We prove that if Σ is a σ -algebra of subsets of a set Ω which is covered by an increasing sequence { Σ n : n ∈ N } of subsets of Σ there exists p ∈ N such that Σ p is a Grothendieck set for ba ( A ) . This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a σ -algebra Σ is covered by an increasing sequence { Σ n : n ∈ N } of subsets, there is p ∈ N such that Σ p is a Nikodým set for ba ( Σ ) . This also reﬁnes the Grothendieck result stating that for each σ -algebra Σ the Banach space (cid:96) ∞ ( Σ ) is a Grothendieck space. Some applications to classic Banach space theory are given.


Introduction
With a different terminology, Valdivia showed in [1] that if a σ-algebra Σ of subsets of a set Ω is covered by an increasing sequence {Σ n : n ∈ N} of subsets, there is p ∈ N such that Σ p is a Nikodým set for ba (Σ).We prove that if Σ is covered by an increasing sequence {Σ n : n ∈ N} of subsets of Σ there is p ∈ N such that Σ p is a Grothendieck set for ba(A) (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 ∞ (Σ) of bounded scalar-valued Σ-measurable functions defined on Ω 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 {Σ n : n ∈ N} is an increasing sequence of subsets of Σ covering Σ, there is p ∈ N such that if {µ n } ∞ n=1 ⊆ ba (Σ) verifies lim n→∞ µ n (A) = 0 for every A ∈ Σ p and {A k : k ∈ N} is a sequence of pairwise disjoint elements of Σ, then lim n→∞ ∑ ∞ k=1 |µ n (A k )| = 0.

Preliminaries
In what follow we use the notation of [2] (Chapter 5).Let R be a ring of subsets of a nonempty set Ω, χ A be the characteristic function of the set A ∈ R and let ∞ 0 (R) = span {χ A : A ∈ R} denote the linear space of all K-valued R-simple functions, K being the scalar field of real or complex numbers.
The dual of ∞ 0 (R) is the Banach space ba(R) of bounded finitely additive scalar-valued measures on R, which we shall assume to be equipped with the variation norm where the supremum is taken over all finite sequences of pairwise disjoint members of R.This is the dual of the supremum-norm • ∞ of ∞ 0 (R).An equivalent norm is given by µ = sup {|µ (A)| : A ∈ R}, which is the dual norm of the gauge • Q .We shall also consider the Banach space ba(R) * equipped with the bidual norm • of • ∞ .The completion of the normed space The Banach space ∞ (R) embeds isometrically into ba (R) * , hence each characteristic function χ A in ∞ 0 (R) with A ∈ R can be considered as a bounded linear functional on ba (R) defined by evaluation χ A , µ = µ (A).So, we may write {χ A : A ∈ R} ⊆ S ba(R) * , where S ba(R) * stands for the unit sphere of ba (R) * , and the set {χ A : A ∈ R}, regarded as a topological subspace of ba (R) * (weak * ), is the same as {χ A : A ∈ R} regarded as a topological subspace of ∞ 0 (R) (weak).A subfamily of an algebra of sets A is called a Nikodým set for ba (A) (cf. [3])if each set {µ α : α ∈ Λ} in ba (A) which is pointwise bounded on is bounded in ba (A), i. e., if sup α∈Λ |µ α (A)| < ∞ for each A ∈ implies that sup α∈Λ |µ α | < ∞.The algebra A is said to have property (N) if the whole family A is a Nikodým set for ba(A).Nikodým's classic boundedness theorem establishes that every σ-algebra has property (N).An algebra A is said to have property (G) if ∞ (A) is a Grothendieck space, i. e., if each weak* convergent sequence in ba(A) is weakly convergent in the Banach space ba(A).The fact that every σ-algebra has property (G) is also due to Grothendieck.Every countable algebra lacks property (N), and the algebra J of Jordan-measurable subsets of the real interval [0, 1] has property (N) but fails property (G) (cf. [4] (Propositions 3.2 and 3.3) and [5]).Let us recall that a sequence {µ n } ∞ n=1 in ba(A) is uniformly exhaustive if for each sequence {A i : i ∈ N} of pairwise disjoint elements of A it holds that lim k→∞ sup n∈N |µ n (A k )| = 0. We shall use the following result, originally stated in [4] (2.3 Definition).Theorem 1.An algebra of sets A has property (G) if and only if every bounded sequence {µ n } ∞ n=1 in ba (A) which converges pointwise on A is uniformly exhaustive.
An algebra A is said to have property (V HS) if every sequence {µ n } ∞ n=1 in ba (A) which converges pointwise on A is uniformly exhaustive.It should be mentioned that (V HS) ⇔ (N) ∧ (G), 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 M of an algebra of sets A will be called a Grothendieck set for ba(A) if each sequence {µ n } ∞ n=1 in ba(A) which is pointwise convergent on M is weakly convergent in ba(A), i. e., if there is µ ∈ ba (A) such that µ n (A) → µ (A) for every A ∈ M then µ n → µ weakly in ba(A).
If an algebra A contains a Grothendieck subset for ba(A), clearly A has property (G).Grothendieck sets are closely related to the so-called Rainwater sets (defined below) for ba(A), and the study of the Rainwater sets for ba(A) 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 {Σ n : n ∈ N} of subsets of Σ there exists p ∈ N such that Σ p is a Grothendieck set for ba(Σ).Indeed, in [1] (Theorem 1) Valdivia showed that if a σ-algebra Σ of subsets of a set Ω is covered by an increasing sequence {Σ n : n ∈ N} of subsets (subfamilies) of Σ, there exists some p ∈ N such that Σ p is a Nikodým set for ba (Σ) or, equivalently, that given an increasing sequence {E n : n ∈ N} of linear subspaces of ∞ 0 (Σ) covering ∞ 0 (Σ), there exists p ∈ 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 σ-algebra Σ is covered by an increasing sequence {Σ n : n ∈ N} of subsets, there exists some p ∈ N such that χ A : A ∈ Σ p , regarded as a subset of the dual unit ball of ba (Σ), is also a Rainwater set for ba (Σ).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).

Rainwater Sets for ba (A)
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 {x n } ∞ n=1 of E that converges pointwise on X, i. e., such that x * x n → x * x for each x * ∈ 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 (X) of real-valued continuous functions over a compact Hausdorff space X equipped with the supremum norm, if K = Ext B C(X) * is the set of the extreme points of the compact subset B C(X) * of C (X) * (weak * ), the Arens-Kelly theorem asserts that K = {± δ x : x ∈ X} (see [13]).By the Lebesgue dominated convergence theorem, if { f n } ∞ n=1 is a norm-bounded sequence in C (X) (with respect to the supremum-norm) then f n → f weakly in C (X) if and only if f n (x) → f (x) for every x ∈ X, that is, f n , µ → f , µ for every µ ∈ C (X) * if and only if f n , δ v → f , δ v for each v ∈ K (see [14] (IV.6.4Corollary)).This is Rainwater's theorem for C (X).In [10] the weak K-analyticity of the Banach space C b (X) 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 (X).The next theorem, based on [3] (Proposition 4.1), exhibits a connection between Rainwater sets and property (G).We include it for future reference and provide a proof for the sake of completeness.Theorem 3. Let A be an algebra of sets.The following are equivalent 1.
A has property (G).

2.
{χ A : A ∈ A} is a Rainwater set for ba (A), considered as a subset of ba (A) * .

3.
The unit ball of ∞ 0 (A) is a Rainwater set for ba (A).

4.
The unit ball of ∞ (A) is a Rainwater set for ba (A).
Proof. 1 ⇒ 2. Assume that A has property (G) and let {µ n } ∞ n=1 be a bounded sequence in ba (A) and µ ∈ ba (A) such that χ A , µ n → χ A , µ for each A ∈ A. i. e., such that µ n (A) → µ (A) for each A ∈ A. By Theorem 1 the sequence M = {µ n : n ∈ N} is (bounded and) uniformly exhaustive on A, so [15] (Corollary 5.2) produces a nonnegative real-valued finitely-additive measure λ on A such that lim λ(E)→0 sup n∈N |µ n (E)| = 0. Hence, [14] (4.9.12 Theorem]) shows that M is relatively weakly sequentially compact.Given that µ n (A) → µ (A) for each A ∈ A, necessarily µ is the only possible weakly adherent point of the sequence {µ n } ∞ n=1 .So we get that µ n → µ weakly in ba (A), which shows that {χ A : A ∈ A} is a Rainwater set for ba (A).
2 ⇒ 3.If B ba(A) * denotes the second dual ball of the closed unit ball B ∞ (A) of ∞ (A) and B 0 stands for the unit ball of ∞ 0 (A), from the relations {χ A : A ∈ A} ⊆ B 0 ⊆ B ba(A) * it follows that B 0 is a also Rainwater set for ba (A).
is a bounded sequence in ba (A).Given that µ n , f → µ, f for every f ∈ B ∞ (A) and given the hypothesis that B ∞ (A) is a Rainwater set for ba (A), we have that µ n → µ weakly in ba (A).
Consequently A has property (G).
Example 1.If Z stands for the algebra generated by the sets of density zero in N, then {χ A : A ∈ Z } is not a Rainwater set for ba (Z ).This follows from the previous theorem and from the fact that Z does not have property (G) (see [16]).
Theorem 4. Assume that A is an algebra of sets.Let M be a Nikodým subset for ba (A) and let {M n : n ∈ N} be an increasing covering of M by subsets of M. If {χ A : A ∈ M} is a Rainwater set for ba (A), there exists some p ∈ N such that χ A : A ∈ M p is a Rainwater set for ba (A).
Proof.Assume that {χ A : A ∈ M} is a Rainwater set for ba (A).First we claim that Let us proceed by contradiction.Assume otherwise that there exists In this case the separation theorem provides µ n ∈ ba (A) So, in particular it holds that Since M is a Nikodým set and {µ n } ∞ n=1 is pointwise bounded on M, it follows that {µ n } ∞ n=1 is bounded in ba (A).So, the fact that µ n (M) → 0 for all M ∈ M along with the assumption that M is a Rainwater set leads to µ n → 0 weakly in ba (A).This is a contradiction, since χ B , µ n = µ n (B) = 1 for every n ∈ N. The claim is proved.
Set Q := {χ A : A ∈ A}.Since we are assuming that M is a Nikodým set for ba (A), the larger set A is also a Nikodým set for ba (A), which implies that ∞ 0 (A) 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 is an increasing sequence of closed absolutely convex sets covering ∞ 0 (A).So, there exists p ∈ N such that is a Rainwater set for ba (A).
We claim that this implies that χ A : A ∈ M p is a Rainwater set for ba (A).In order to establish the claim it suffices to show that abx χ A : A ∈ M p is a Rainwater set for ba (A).So, let {λ n } ∞ n=1 be a bounded sequence in ba (A) such that u, λ n → 0 for every u ∈ abx χ • ∞ is a Rainwater set for ba (A), we get that λ n → 0 weakly in ba (A).Therefore the absolutely convex set abx χ A : A ∈ M p is a Rainwater set for ba (A), a stated.

Corollary 1.
Let A be an algebra of sets with property (V HS).If {A n : n ∈ N} is an increasing covering of A consisting of subsets of A, there is some p ∈ N such that χ A : A ∈ A p is a Rainwater set for ba (A).
Proof.This is a straightforward consequence of the Theorem 4 for M = A, since as mentioned earlier an algebra A has property (V HS) if and only if A has both properties (N) and (G) (this also can be found in [7] (Theorem 4.2)).So, on the one hand A is a Nikodým set for ba (A) and, on the other hand, according to Theorem 3, the family {χ A : A ∈ A} is a Rainwater set for ba (A).
Proof of Theorem 2. If Σ is a σ-algebra of subsets of a set Ω which is covered by an increasing sequence {Σ n : n ∈ N} of subsets of Σ, Corollary 1 and Valdivia's result [1] provide an index p ∈ N such that Σ p is a Nikodým set for ba (Σ) at the same time that χ A : A ∈ Σ p is a Rainwater set for ba (Σ).
If {µ n } ∞ n=1 verifies that µ n (A) → µ (A) for every A ∈ Σ p , the sequence {µ n } ∞ n=1 is bounded in ba (Σ) since Σ p is a Nikodým set for ba (Σ).But then µ n → µ weakly in ba (Σ) due to χ A : A ∈ Σ p is a Rainwater set for ba (Σ).Consequently Σ p is a Grothendieck for ba (Σ) and we are done.
Corollary 2. If {Λ n : n ∈ N} is an increasing sequence of subsets of Σ = 2 N covering 2 N , there exists some p ∈ N such that each sequence {µ n } ∞ n=1 in ba 2 N that converges pointwise on Λ p converges weakly in ba 2 N = * ∞ .
We complete our study of Rainwater sets for ba (A) with the following result.Note that if X w * (weak* closure) with X ⊆ B ba(A) * is a Rainwater set for ba (A) then X could not be a Rainwater set for ba (A).However the following property holds.
Theorem 5. Let A be an algebra of sets.Assume that {χ A : A ∈ A} is a Grothendieck set for ba (A).
If {χ A : A ∈ M} is a G δ -dense subset of {χ A : A ∈ A} under the relative weak* topology of ba (A) * or, which is the same, under the relative weak topology of ∞ 0 (A), then {χ A : A ∈ M} is a Grothendieck set for ba (A).
G n is a nonempty intersection of countably many zero-sets of {χ A : A ∈ A}, hence a non-empty G δ -set in {χ A : A ∈ A} in the relative weak topology of ∞ 0 (A).According to the hypothesis G meets {χ A : A ∈ M}.Hence there exists M B ∈ M such that χ M B ∈ G ∩ {χ A : A ∈ M}, which means that µ n (M B ) = µ n (B) for every n ∈ N. Since µ n (M B ) → 0, it follows that µ n (B) → 0. So, we conclude that µ n (B) → 0 for every B ∈ A. Putting together that (i) {χ A : A ∈ A} is a Grothendieck set for ba (A), and (ii) µ n (B) → 0 for all B ∈ A, we get that µ n → 0 weakly in ba (A).Thus {χ A : A ∈ M} is a Grothendieck set for ba (A).

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 (Σ), Nikodým's pointwise convergence theorem in ca (Σ) and the usual characterization of weak convergence in ca (Σ), the linear subspace of ba (Σ) consisting of the countably additive measures in Σ (see [18] (Chapter 7)).Proposition 1.Let Σ be a σ-algebra of subsets of a set Ω.If {Σ n : n ∈ N} is an increasing sequence of subsets of Σ covering Σ, there exists some p ∈ N enjoying the following property.If {µ n } ∞ n=1 ⊆ ba (Σ) verifies lim n→∞ µ n (A) = 0 for every A ∈ Σ p and {A k : k ∈ N} is a sequence of pairwise disjoint elements of Σ, then Proof.According to Theorem 2 there is p ∈ N such that Σ p is Grothendieck set for ba (Σ).So, if lim n→∞ µ n (A) = 0 for every A ∈ Σ p , then µ n → 0 weakly in ba (Σ).In particular, µ n (A) → 0 for every A ∈ Σ.Hence, (1) holds by Phillip's classic theorem.
Proposition 2. Let Σ be a σ-algebra of subsets of a set Ω.If {Σ n : n ∈ N} is an increasing sequence of subsets of Σ covering Σ, there exists some p ∈ N such that if {µ n } ∞ n=1 ⊆ ca (Σ) verifies that µ n (A) → µ (A) for every A ∈ Σ p then the set {µ n : n ∈ N} is uniformly exhaustive and µ ∈ ca (Σ).Proposition 3. Let Σ be a σ-algebra of subsets of a set Ω.If {Σ n : n ∈ N} is an increasing sequence of subsets of Σ covering Σ, there exists some p ∈ N such that µ n → µ weakly in ca (Σ) if and only if µ n (A) → µ (A) for every A ∈ Σ p .