This is a reprint of the paper On exhaustive vector measures

by Juan Carlos Ferrando & M. López-Pellicer

published in Rev. Colombiana Mat. 28 (1994), pp. 13–21

ON EXHAUSTIVE VECTOR MEASURES

Juan Carlos Ferrando Manuel López–Pellicer

Universidad Politécnica, Valencia, ESPAÑA

Abstract. Certain conditions related with the De Wilde and Valdivia closed graph theorems enable us to guarantee the exhaustivity of certain bounded vector measures. To obtain some of these conditions we show incidentally that the bounded countably valued scalar function space with the supremum norm is ultrabornological. Keywords and phrases. Closed graph theorems, spaces Γ_{r} and Λ_{r}, barrelled spaces, finitely (countably) additive vector measure, bounded vector measure, exhaustive vector measure 1991 Mathematics Subject Classification. 46A03, 46G10.

In this paper ‘space’ means ‘locally convex Hausdorff space over the real or complex field’. Unless other thing is stated, by ‘topology’ we understand ‘Hausdorff locally convex topology’.

If s is a positive integer, then the countable family of subspaces W = {X_{m_{1}m_{2}···m_{p}} | m_{r} ∈ ℕ | 1 ≤ r ≤ p ≤ s} of a space X is an s–net if {X_{m_{1}} | m_{1} ∈ ℕ} is an increasing covering of X and the sequence {X_{m_{1}m_{2}···m_{j}} | m_{j} ∈ ℕ} is an increasing covering of X_{m_{1}m_{2}···m_{j−1}}, for 2 ≤ j ≤ s. We write W_{s} = {X_{m_{1}m_{2}...m_{s}} | m_{r} ∈ ℕ, 1 ≤ r ≤ s}. A space X is Γ_{r}, [11] (Λ_{r}, [12]) if given any quasi-complete (locally complete) subspace G of X^{∗}(σ(X^{∗}, X)), X^{∗} being the algebraic dual of X, such that G meets the topological dual X′ of X in a weak* dense subspace of X′, then G contains X′.

Typeset by AMS-TEX

This paper has been partially supported by DGICYT, PB91-0407, and IVEI (Institució Valenciana d’Estudis i Investigació), proyecto 003/023

B_{r}-complete spaces are Γ_{r}, and reflexive Banach spaces and Fréchet-Schwartz spaces provide some examples of Λ_{r}-spaces.

By Σ we shall denote a σ-algebra of subsets of the set Ω. If μ is a mapping from Σ in a space X such that μ(A ∪ B) = μ(A) + μ(B) for each disjoint pair (A, B) of elements of Σ, then μ is said to be a finite additive vector measure or simply a vector measure.

A vector measure μ is bounded if the set {μ(A) | A ∈ Σ} is bounded, and μ is called exhaustive if whenever we have a sequence {E_{i} | i ∈ ℕ} of pairwise disjoint elements of Σ, then lim μ(E_{i}) = 0. It is well-known that each exhaustive vector measure is bounded, and the aim of this paper is to find out some conditions implying that bounded measures are exhaustive. A vector measure μ is strongly additive if for every sequence {E_{i} : i ∈ ℕ} of pairwise disjoint elements of Σ, ∑{μ(E_{i}) | i ∈ ℕ} converges in X. If, additionally, ∑{μ(E_{i}) | i ∈ ℕ} = μ (∪{E_{i} | i ∈ ℕ}), the measure μ is called countably additive. Obviously, each strongly additive measure is exhaustive, the converse being true when the space X is sequentially complete.

In the space L^{∞}(Ω, Σ) of all bounded Σ-measurable scalar functions endowed with the supremum norm, we are going to consider the following subspaces:

• The subspaces S_{c}(Ω, Σ) of the functions of countable range. Note that a function f ∈ S_{c}(Ω, Σ) is determined by the bounded sequence of scalars {a_{n} | n ∈ ℕ} formed by the elements of f(Ω) and the sets f^{−1}(a_{n}).

• The subspace L(A) of S_{c}(Ω, Σ) formed by those functions determined by all the elements of ℓ^{∞} and the sequence A = {A_{n} | n ∈ ℕ} of pairwise disjoint elementes of Σ. Obviously L(A) is isometric to ℓ^{∞}.

• The linear span of the characteristic funtions e(E) of all E ∈ Σ. This space

will be denoted either by S(Ω, Σ) or by ℓ_{0}^{∞}(Ω, Σ). When Ω is the set ℕ of natural numbers and Ω is the σ-algebra of all subsets of ℕ, we shall write ℓ_{0}^{∞} instead of ℓ_{0}^{∞}(ℕ, P(ℕ)).

If μ : Σ → X is a vector measure, the linear mapping S from S(Ω, Σ) into X such that S(e(E)) = μ(E) for each E ∈ Σ will be called the linear mapping associated to μ. If μ is bounded, we have that S is continuous. In fact, if |λ_{1}|+|λ_{2}|+…+|λ_{n}| ≤ 1, E_{1}, E_{2}, …, E_{n} ∈ Σ and p is a continuous seminorm on X we have that

p(S(λ_{1}e(E_{1}) + λ_{2}e(E_{2}) + … + λ_{n}e(E_{n}))) ≤ sup{p(μ(E)) | E ∈ Σ}

and continuity of S follows immediately from this inequality and the following result given by M. Valdivia in [13], Lemmas 1 and 2:

(a) if B stands for the closed unit ball of ℓ_{0}^{∞}(Ω, Σ), then the absolutely convex hull of \{e(E) | E ∈ Σ\} contains \frac{1}{4}B.

The next result is an extension, due to L. Drewnowski, [4] and [5], of a well known result of H. P. Rosenthal for Banach spaces [10].

(1) ”(b)” Let T be a linear continuous mapping from ℓ^{∞} into a Hausdorff topological vector space X and let {e_{n} | n ∈ ℕ} be the sequence of the unit vectors of ℓ^{∞}. If X has a neighbourhood of the origin U such that the set {n ∈ ℕ | T e_{n} ∉ U} is infinite, then X contains a copy of ℓ^{∞}.

We use this in order to prove our first theorem (see also [8], Theorem 1, Corollary A).

Theorem 1. Let μ be a bounded vector measure defined in Σ with values in a sequentially complete topological vector space X. If X does not contain any copy of ℓ^{∞}, then μ is strongly additive.

Proof. We know that the linear mapping S : S(Ω, Σ) → X associated with μ is continuous. The metrizability of L^{∞}(Ω, Σ) and the sequential completeness of X enables us to extend S to a continuous linear mapping \overline{S} : L^{∞}(Ω, Σ) → X.

If {A_{i} | i ∈ ℕ} is a sequence of pairwise disjoint elements of Σ, A_{0} = Σ ∖ ∪{A_{i} | i ∈ ℕ}, A is the sequence {A_{0}, A_{1}, A_{2}, … } and T is the restriction of \overline{S} to L(A) then, by property b), the sequence (S(e(A_{i}))) converges to zero given that X does not contain a copy of ℓ^{∞}. Therefore the measure μ is exhaustive and given that X is sequentially complete, μ is strongly additive. □

Corollary 1. Let μ : Σ → X be a bounded measure and suppose that X is sequentially complete. If f : X → Y is a linear mapping with closed graph taking its values into a webbed space Y which does not contain a copy of ℓ^{∞}, then fμ is exhaustive.

Proof. Let \overline{S} : L^{∞}(Ω, Σ) → X be the continuous linear mapping determined in Theorem 1. Then f\overline{S} : L^{∞}(Ω, Σ) → Y has a closed graph and, according to the De Wilde closed graph theorem [2], f\overline{S} : L^{∞}(Ω, Σ) → Y is continuous. Then, the conclusion is obtained as in Theorem 1, considering the restriction of f\overline{S} to L(A). □

In Theorem 4 of [13], M. Valdivia has obtained the following result:

(c) Let μ be a bounded finite additive X-valued vector measure and let {F_{n} | n ∈ ℕ} be an increasing sequence of Γ_{r}-spaces covering a space F. If f is a linear mapping defined in X with values in F which has closed graph, then there is a positive integer q such that fμ is a F_{q}-valued bounded finite additive measure on Σ.

In [9], Theorem 16, B. Rodríguez-Salinas gives the following extension of result (c):

(d) Let μ : Σ → X be a bounded finite additive measure and let F be a space which has an increasing covering {F_{n} | n ∈ ℕ} of subspaces such that for each n there is a topology τ_{n} on F_{n}, finer than that induced by F, under which F_{n}(τ_{n}) is a sequentially complete Γ_{r}-space which does not contain a copy of ℓ^{∞}. If f : X → F is a linear mapping with closed graph, then there is a positive integer n such that fμ : Σ → F_{n}(τ_{n}) is a strongly additive measure.

Our next result follows from these two results and from [6], Theorem 1, as well as from the observation that in result (d) there is in F a topology τ_{F} weaker than the initial one such that f : X → F (τ_{F}) is continuous and the map fμ : Σ → F (τ_{F}) is a bounded finite additive measure.

Theorem 2. Let μ : Σ → X be a bounded measure defined in a space X containing an s-net W such that every H ∈ W_{s} has a topology τ_{H} finer than the induced by the original topology of X and such that H(τ_{H}) is a Γ_{r}-space which does not contain any copy of ℓ^{∞}. Then there is a G ∈ W_{s} such that the mapping fμ : Σ → G(τ_{G}) is exhaustive.

Proof. Let S : S(Ω, Σ) → X be the continuous linear mapping associated with μ. By Theorem 1 of [6] there is a G ∈ W_{s} such that S^{−1}(G) is a dense and barrelled subspace of S(Ω, Σ). But according to Theorems 1 and 14 of [11] the restriction of S to S^{−1}(G) has a continuous extension \overline{S} : L^{∞}(Ω, Σ) → G(τ_{G}) which agrees with S in S(Ω, Σ). Now the conclusion follows exactly as in Theorem 1, changing X by G(τ_{G}). □

If G(τ_{G}) is sequentially complete then obviously fμ : Σ → G(τ_{G}) is strongly additive.

Corollary 2. Let μ : Σ → X be a bounded measure defined in a space X that contains an s-net W such that every H ∈ W_{s} has a topology τ_{H} finer than the induced by the original topology of X and such that H(τ_{H}) is a Γ_{r}-space. If f is a linear mapping with closed graph from X into a webbed space Y which does not contain a copy of ℓ^{∞}, then fμ is exhaustive.

Proof. In Theorem 2 we have obtained the continuous linear mapping \overline{S} : L^{∞}(Ω, Σ) → G(τ_{G}). Then, f\overline{S} : L^{∞}(Ω, Σ) → Y has closed graph and is continuous according to De Wilde closed graph theorem. The conclusion follows as in Theorem 1, T being now the restriction of f\overline{S} to L(A). □

If the webbed space is sequentially complete then fμ is strongly additive.

Examples.

1. Let E be a Banach space having a separable strong dual E′(β(E′, E)) and let μ : Σ → E′(σ(E′, E)) be a bounded measure. E′(β(E′, E)) is a webbed space and also a Γ_{r}–space and since it is separable, it cannot contain any copy of ℓ^{∞}. Clearly, E′(σ(E′, E)) is also a Γ_{r}-space. Hence, if μ : Σ → E′(σ(E′, E)) is a bounded measure, the preceding corollary and the completeness of E′(β(E′, E)) imply that μ : Σ → E′(β(E′, E)) is strongly additive.

2. Let now E be a Fréchet space such that E′(β(E′, E)) does not contain a copy of ℓ^{∞}. Let {V_{n}, n ∈ ℕ} be a fundamental system of neighbourhoods of zero in E, and let E_{n}(τ_{n}) denote the Banach space generated by V_{n}^{0}, with its Minkowski functional as norm. The topology τ_{n} is finer than the induced by σ(E′, E) and E_{n}(τ_{n}) is a Γ_{r}-space. The immersion i : E′(σ(E′, E)) → E′(β(E′, E)) has obviously closed graph. Now, if the measure μ : Σ → E′(σ(E′, E)) is bounded, Corollary 2 implies that μ : Σ → E′(β(E′, E)) is strongly additive.

Our aim now is to debilitate the conditions imposed to X in the last Corollary. To compensate we have to strengthen the conditions imposed to the bounded vector measure μ.

Given a continuous seminorm p on the space X. The p-variation |μ|_{p} of the vector measure μ with values in X is defined by

|μ|_{p}(E) = sup {∑_{i=1}^{n} p(μ(E_{i})) | {E_{1}, E_{2}, …, E_{n}} ∈ P(E)} ,

where P(E) denotes the family of all finite partitions of E by elements of Σ. The vector measure μ is said of bounded variation if |μ|_{p}(Ω) < ∞ for every continuous seminorm p of X. If the space X is sequentially complete and μ is a X-valued vector measure of bounded variation, then μ is strongly additive because Σ{p(μ(E_{i})) | i ∈ ℕ} ≤ |μ|_{p}(Ω). There are countably additive vector measures which are not of bounded variation ([3], pp. 7-8).

Lemma. S_{c}(Ω, Σ) is ultrabornological.

Proof. Let P_{c} be the family of all countable partitions A of Ω formed by elements of Σ. We are going to prove that S_{c}(Ω, Σ) is the locally convex hull of the family {L(A) | A ∈ P_{c}}. Let V be an absolutely convex subset of S_{c}(Ω, Σ) such that V ∩ L(A) is a neighbourhood of zero in L(A) for every A ∈ P_{c}. In order to prove that V is a neighbourhood of the origin in S_{c}(Ω, Σ), one can easily show that it suffices, because of result (b), to prove that there exists a λ > 0 such that λe(E) ⊂ V, for every E ∈ Σ.

Suppose that this property is not true, and let n_{1} ∈ ℕ be such that e(Ω) ∈ (n_{1}/6)V. Then there must be some A_{1} ∈ Σ such that e(A_{1}) ∉ (4n_{1}/3)V. But, since e(A_{1}) = e(Ω) − e(Ω ∖ A_{1}), it follows that e(Ω ∖ A_{1}) ∉ n_{1}V. So we have shown that e(A_{1}) ∉ n_{1}V and e(Ω ∖ A_{1}) ∉ n_{1}V.

Let n_{2} > 2n_{1} be such that e(A_{1}) ∈ (n_{2}/12)V. Then

e(Ω ∖ A_{1}) = e(Ω) − e(A_{1}) ∈ (n_{1}/6)V + (n_{2}/12)V ⊂ (n_{2}/6)V.

So we have obtained that e(A_{1}) ∈ (n_{2}/6)V and e(Ω ∖ A_{1}) ∈ (n_{2}/6)V.

Since e(E) = e(E ∩A_{1})+e(E ∩(Ω∖A_{1})) for every E ∈ Σ, we deduce that either V does not absorb the family {e(E) | E ⊂ A_{1}, E ∈ Σ} or it does not absorb the family {e(E) | E ⊂ Ω ∖ A_{1}, E ∈ Σ}.

We may suppose that V does not absorb {e(E) | E ⊂ Ω ∖ A_{1}, E ∈ Σ}. In this moment we have e(A_{1}) ∉ n_{1}V, e(Ω ∖ A_{1}) ∈ (n_{2}/6)V and there is in Ω ∖ A_{1} a subset A_{2} ∈ Σ such that e(A_{2}) ∉ (4n_{2}/3)V. We can obviously repeat the preceding argument, and proceeding by induction we can obtain a sequence A = {A_{n} | n ∈ ℕ} of pairwise disjoint subsets of Σ and a sequence n_{1} < n_{2} < ⋯ of positive integer numbers such that e(A_{j}) ∉ n_{j}V, contradicting that L(A) ∩ V is a neighbourhood of zero in L(A). □

Theorem 3. Let μ : Σ → X be a countably additive measure of bounded variation and suppose that every series in the space X that is subseries convergent is bounded multiplier convergent. Let Y be a webbed space that does not contain any copy of ℓ^{∞} and let f be a linear mapping from X into Y with closed graph. Then fμ is an exhaustive vector measure.

Proof. Since μ is countably additive we have that, given a sequence {E_{n} | n ∈ ℕ} of pairwise disjoint elements of Σ, the series ∑{μ(E_{i}) | i ∈ ℕ} is subseries convergent and, consequently, by the hypotheses imposed on the space X, the series ∑{μ(E_{i}) | i ∈ ℕ} is bounded multiplier convergent. Then for every {a_{n} | n ∈ ℕ} ∈ ℓ^{∞}, the series ∑{a_{i}μ(E_{i}) | i ∈ ℕ} is also subseries convergent.

This obsevation enables us to define a linear mapping U from S_{c}(Ω, Σ) into X by Ug = ∑{a_{i}μ(E_{i}) | i ∈ ℕ}, where {a_{n} | n ∈ ℕ} ∈ ℓ^{∞} and g(E_{i}) = {a_{i}} for every i ∈ ℕ. The definition is correct because if {F_{m} | m ∈ ℕ} is a refinement of {E_{n} | n ∈ ℕ} we have by the subseries convergence that

∑{b_{i}μ(F_{i}) | i ∈ ℕ} = ∑{a_{i}μ(E_{i}) | i ∈ ℕ},

since in the series ∑{b_{i}μ(F_{i}) | i ∈ ℕ} we may reorder and associate as needed. The same argument shows that U is linear.

The mapping U is continuous since taking any continuous seminorm p we have that

p(Ug) = p(∑{a_{i}μ(E_{i}) | i ∈ ℕ}) ≤ |μ|_{p}(Ω) × ∥{a_{i} | i ∈ ℕ}∥_{∞} = |μ|_{p}(Ω) × ∥g∥

Therefore T = fU is a mapping with closed graph from S_{c}(Ω, Σ) into Y. Since S_{c}(Ω, Σ) is ultrabornological according to the previous lemma, then T is continuous by De Wilde closed graph theorem [2].

Finally, to prove that fμ is exhaustive, let {A_{i} | i ∈ ℕ} be a sequence of pairwise disjoint elements of Σ, A_{0} := Ω ∖ ∪{A_{i} | i ∈ ℕ} and A = {A_{0}, A_{1}, A_{2}, … }. Since L(A) is isometric to ℓ^{∞}, we have that 0 = lim T (e(A_{n})) = lim fμ(A_{n}). □

In the preceding theorem the space X has to verify that every subseries convergent series is bounded multiplier convergent or, briefly, BM–convergent. The next proposition gives a sufficient condition which implies this property.

Proposition 1. Let ∑{x_{n} | n ∈ ℕ} be subseries convergent in a space X containing an s-net W such that every F ∈ W_{s} has a finer topology τ_{F} such that F (τ_{F}) is a Λ_{r}-space. Then there is a G ∈ W_{s} such that ∑{x_{n} | n ∈ ℕ} is a BM-convergent series in G(τ_{G}).

Proof. The subseries convergence condition implies that the mapping K from ℓ_{0}^{∞}(σ(ℓ_{0}^{∞}, ℓ^{1})) into X given by K(e(A)) = ∑{x_{n} | n ∈ A} is well-defined and linear. It is also continuous since given an x^{∗} ∈ X′ we have that

(K^{t}x^{∗})e(A) = x^{∗}(∑{x_{n} | n ∈ A}) = ∑{x^{∗}x_{n} | n ∈ A}
= ⟨(x^{∗}x_{n}, n ∈ ℕ), e(A)⟩

and therefore the sequence (x^{∗}x_{n}) is subseries convergent, which implies that (x^{∗}x_{n}) ∈ ℓ^{1}.

By [6], Theorem 1, there is a G ∈ W_{s} such that H = K^{−1}(G) is a barrelled and dense subspace of ℓ_{0}^{∞}. Then if M is a subset of ℓ^{1} which is σ(ℓ^{1}, H) bounded, M is ℓ_{0}^{∞}-equicontinuous and, since H is dense in ℓ^{∞}, it is also ℓ^{∞}-equicontinuous. Thus we have that every u ∈ ℓ^{∞} is bounded on M and hence ℓ^{∞} is contained in the bounded closure of H respect to the dual pair ⟨H, ℓ^{1}⟩. From Theorems 2 and 6 of [12], it follows that the restriction of K to H possesses a continuous linear extension T : ℓ^{∞}(σ(ℓ^{∞}, ℓ^{1})) → G(σ(G, G(τ_{G})′)). Now, the continuity of K implies that T is also an extension of K.

Given a vector a = {a_{n} | n ∈ ℕ} ∈ ℓ^{∞} we write a^{(n)} = (a_{1}, a_{2}, …, a_{n−1}, a_{n}, 0, 0, 0, …) for every n ∈ ℕ. Then lim a^{(n)} = a in σ(ℓ^{∞}, ℓ^{1}) and therefore ∑{a_{n}x_{n} | n ∈ ℕ} is σ(G, G(τ_{G})′)–convergent in G. Changing some of the a_{n}’s by zeros we obtain that ∑{a_{n}x_{n} | n ∈ ℕ} is weakly subseries convergent, and then the conclusion follows directly from the Orlicz–Pettis theorem. □

From the two former results we obtain the following corollary.

Corollary. Let μ : Σ → X be a countably additive measure with bounded variation and suppose that the space X contains an s-net W such that every F ∈ W_{s} has a finer topology τ_{F} under which F (τ_{F}) is a Λ_{r}-space. Suppose that Y is a webbed space that does not contain a copy of ℓ^{∞}. If f is a linear mapping from X into Y with closed graph, then fμ is an exhaustive vector measure.

If X and Y are two spaces, a linear mapping f : X → Y is said to be subcontinuous if given any series ∑{x_{n} | n ∈ ℕ} which is subseries convergent, then ∑{f(x_{n}) | n ∈ ℕ} converges to f (∑{x_{n} | n ∈ ℕ}).

In [9], B. Rodríguez–Salinas proves the following, which generalizes Theorem 1 in G. Bennett and N. J. Kalton [1].

(e) Let X and Y be two spaces. Let us suppose that Y is a sequentially complete Γ_{r}(ℓ_{0}^{∞})-space. If f : X → Y is a linear mapping with closed graph and Y does not contain a copy of ℓ^{∞}, then f is subcontinuous.

This result has motivated our following theorem.

Theorem 4. Let X and Y be two spaces. Let W be an s-net in Y such that every L ∈ W_{s} has a finer topology τ_{L} such that L(τ_{L}) is a Γ_{r}-space sequentially complete which does not contain a copy of ℓ^{∞}. If f : X → Y is a linear mapping with closed graph, then f is subcontinuous.

Proof. Let us assume that ∑{x_{n} | n ∈ ℕ} is subseries convergent in X. In [9], Theorem 14, it is proved that the mapping K : ℓ_{0}^{∞} → X defined by

K({a_{n} | n ∈ ℕ}) = ∑{a_{n}x_{n} | n ∈ ℕ}

is continuous. In fact, if sup {|a_{n}|, n ∈ ℕ} ≤ 1 and y′ ∈ Y′, then

|y′(K({a_{n} | n ∈ ℕ}))| < ∞,

since ∑{y′x_{n} | n ∈ ℕ} is absolutely convergent.

Consequently by [6], Theorem 1, there is a G ∈ W_{s} such that (fK)^{−1}(G) is barrelled and dense in ℓ_{0}^{∞}.

By [11], Theorems 1 and 14, there is a continuous extension U : ℓ^{∞} → G(τ_{G}) of the restriction of fK to (fK)^{−1}(G). Since f has closed graph, U coincides with fK in ℓ_{0}^{∞}. Then, exactly as we did in Theorem 1, but working with the restriction of U to L(A) instead of T, we obtain that the G(τ_{G})–valued bounded measure μ defined in the σ-algebra of subsets of ℕ by μ(A) = U(e(A)) is strongly additive. Hence, taking A_{n} = {n}, we have that

∑{μ({n}) | n ∈ ℕ} = ∑{f(x_{n}) | n ∈ ℕ}

converges in G(τ_{G}) and hence in Y. Thus by the closed graph condition,

∑{f(x_{n}), n ∈ ℕ} = f(∑{x_{n} | n ∈ ℕ}).

Acknowledgment

This paper was suggested by Professor B. Rodríguez–Salinas reference [9].

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(Recibido en septiembre de 1993)

Juan Carlos Ferrando

Departamento de Matemática Aplicada

Escuela Universitaria de Informática, Universidad Politécnica

Apartado 22012 (E–46071), Valencia, ESPAÑA e–mail: jcferran@mat.upv.es Manuel López–Pellicer

Departamento de Matemática Aplicada

Escuela Superior de Ingenieros Agrónomos, Universidad Politécnica

Apartado 22012 (E–46071), Valencia, ESPAÑA e–mail: mlopezp@mat.upv.es