Absolutely continuous operators on function spaces and vector measures

Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L0(μ) such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^\infty(\mu) \subset E \subset L^1(\mu)}$$\end{document}. We study absolutely continuous linear operators from E to a locally convex Hausdorff space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(X, \xi)}$$\end{document}. Moreover, we examine the relationships between μ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators Tm : L∞(μ) → X. In particular, we characterize relatively compact sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}}$$\end{document} in caμ(Σ, X) (= the space of all μ-absolutely continuous measures m : Σ → X) for the topology \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{T}_s}$$\end{document} of simple convergence in terms of the topological properties of the corresponding set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{T_m : m \in \mathcal{M}\}}$$\end{document} of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L∞(μ) → X.


Introduction and terminology
For terminology concerning vector lattices and function spaces we refer the reader to [1], [2], [10]. Throughout the paper we assume that ( , , μ) is a complete finite atomless measure space and L 0 (μ) denotes the corresponding space of μ-equivalence classes of all -measurable real valued-functions defined on . Let E be an M. Nowak (B) Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Szafrana 4A, 65-516 Zielona Gora, Poland e-mail: M.Nowak@wmie.uz.zgora.pl ideal of L 0 (μ) such that L ∞ (μ) ⊂ E ⊂ L 1 (μ), and let E ∼ and E ∼ n stand for the order dual and order continuous dual of E respectively. Then E ∼ n separates the points of E and it can be identified with the Köthe dual E of E through the mapping E v → ϕ v ∈ E ∼ n , where ϕ v (u) = uvdμ for all u ∈ E. It is known that the Mackey topology τ (E, E ∼ n )(= τ (E, E )) is a locally solid Lebesgue topology. The so-called order-bounded topology τ 0 can be defined on E as the finest locally convex topology on E for which every order interval in E is a bounded set (see [11]). A local base B 0 at zero for τ 0 is the class of all absolutely convex subsets of E that absorb all order bounded sets in E. Then τ 0 coincides with the Mackey topology τ (E, E ∼ ). Note that if u n , u ∈ E and u n → u uniformly on , then u n → u for τ 0 .
From now on we assume that (X, ξ) is a locally convex Hausdorff space (for short, lcHs) and let P ξ denote the set of all ξ -continuous seminorms on X. By X ξ we denote the topological dual of (X, ξ). We denote by σ (L , K ) and τ (L , K ) the weak topology and the Mackey topology on L with respect to a dual pair L , K .
Recall that a linear operator T : E → X is said to be order-bounded (resp. orderweakly compact), if for each u ∈ E + , the set T ( [−u, u]) is ξ -bounded (resp. relatively σ (X, X ξ )-compact) in X (see [6] Following [13] a linear operator T : E → X is said to be absolutely continuous if for each u ∈ E, T (1 A n u) → 0 for ξ whenever μ(A n ) → 0, (A n ) ⊂ . Absolutely continuous operators on Orlicz spaces and Frechét function spaces have been examined by Orlicz and Wnuk (see [12,13]).
In Sect. 2 we study absolutely continuous operators T : E → X. We show that a linear operator T : E → X is absolutely continuous if and only if T is (τ (E, E ∼ n ), ξ )continuous. We characterize relatively compact sets in the space L τ,ξ (E, X ) of all (τ (E, E ∼ n ), ξ )-continuous linear operators T : E → X, provided with the topology of simple convergence. In Sect. 3 we examine the relationships between μ-absolutely continuous vector measures m : → X and the corresponding integration operators T m : L ∞ (μ) → X.

Absolutely continuous operators on function spaces
We start with the following result.

Proposition 2.1 Assume that T : E → X is an absolutely continuous linear operator.
Then T is (τ 0 , ξ)-continuous.
Proof In view of Proposition 1.1 it is sufficient to show that T ( [−u, u]) is ξ -bounded in X for every u ∈ E + . For this purpose one can repeat the proof of Theorem 1 in [13]. Now we present a characterization of absolutely continuous operators on E.

Proposition 2.2
For a linear operator T : E → X the following statements are equivalent: (iv) ⇒(v) It is obvious.
(v)⇐⇒(vi) It is enough to repeat the reasoning in the proof of Proposition 4 in [13] and use Proposition 2.1 and the fact that u n → 0 in E for τ 0 whenever u n → 0 uniformly on .
, ξ )-continuous linear operators from E to X, equipped with the topology T s of simple convergence. Let P ξ be the family of all ξ -continuous seminorms on X. Then T s is generated by the family {q p,u : p ∈ P ξ , u ∈ E} of seminorms, where q p,u (T ) = p(T (u)) for all T ∈ L τ,ξ (E, X ).
The following result will be of importance (see [15,Theorem 2]).
n . Now we can state a characterization of relative T s -compactness in L τ,ξ (E, X ). X ). Then the following statements are equivalent: Let W be an absolutely convex and ξ -closed neighbourhood of 0 for ξ in X. Then the polar W 0 of W (with respect to the dual pair E, E ξ ), is a σ (X ξ , X )-closed and ξ -equicontinuous subset of X ξ (see [1,Theorem 9.21]). Then by Theorem 2.4 the set H = {x • T :

Corollary 2.6
Assume that K is a relatively T s -compact subset of L τ,ξ (E, X ). Then K is uniformly μ-absolutely continuous, i.e., for each u ∈ E and p ∈ P ξ we have Proof In view of Theorem 2.4, K is (τ (E, E ∼ n ), ξ )-equicontinuous. Let p ∈ P ξ and ε > 0 be given. Then there exists a τ (E, E ∼ n )-neigbourhood V of 0 in E such that for each T ∈ K we have p(T (u)) ≤ ε for all u ∈ V. Let u ∈ E and μ(A n ) → 0 and let u n = 1 A n u for n ∈ N. Note that u n → 0(μ) and |u n (ω)| ≤ |u(ω)|μ-a.e. for all n ∈ N. Hence by the Riesz theorem for every subsequence (u k n ) of (u n ) there exists a subsequence (u l kn ) of (u k n ) such that u l kn (ω) → 0μ-a.e. This means that u l kn is a Lebesgue topology. It follows that u n → 0 for τ (E, E ∼ n ). Then there exists n ε ∈ N such u n ∈ V for n ≥ n ε , and hence sup T ∈K p(T (1 A n u)) ≤ ε for n ≥ n ε .

Absolutely continuous vector measures
Let (X, ξ) be a quasicomplete lcHs and m : → X be a ξ -bounded vector measure (i.e., the range of m is ξ -bounded in X ) and m(A) = 0 if μ(A) = 0, A ∈ (in symbols, m μ).
Then the integral udm is well defined and the corresponding integration operator T m : L ∞ (μ) → X given by T m (u) = udm is ( · ∞ , ξ)-continuous and linear, and for each x ∈ X ξ , (see [9], [14,Lemma 6]). Conversely, let T : L ∞ (μ) → X be a ( · ∞ , ξ)-continuous linear operator, and let m(A) = T (1 A ) for A ∈ . Then m : → X is a ξ -bounded vector measure such that m μ (called the representing measure of T ) and T m (u) = T (u) for all u ∈ L ∞ (μ).
An important example of a quasicomplete lcHs is the space L(Y, Z ) of all bounded linear operators between Banach spaces Y and Z , provided with the strong operator topology.
Recall that a vector measure m : → X is said to be μ-absolutely continuous m(A n ) → 0 for ξ whenever μ(A n ) → 0, (A n ) ⊂ (see [5,Definition 3,p. 11]). Now we characterize μ-absolutely continuous measures in terms of the properties of the corresponding integration operators.

Proposition 3.1 Assume that (X, ξ) is a quasicomplete lcHs. Let m :
→ X be a ξ -bounded vector measure such that m μ. Then the following statements are equivalent: and this means that x • T m ∈ L ∞ (μ) ∼ n .
As a consequence of Proposition 3.1 we get the following Pettis type theorem for countably additive measures (see [5, Theorem 1, p. 10]). that (X, ξ) is a quasicomplete lcHs. Let m : → X be a ξ -countably additive measure. Then the following statements are equivalent:

Corollary 3.2 Assume
Let ca( , X ) stand for the space of all ξ -countably additive measures m : → X. By ca μ ( , X ) we denote the subspace of ca( , X ) consisting of all m ∈ ca( , X ) that are μ-absolutely continuous. Denote by T s the topology of simple convergence in ca( , X ). Then T s is generated by the family {q p,A : p ∈ P ξ , A ∈ } of seminorms, where for all m ∈ ca( , X ).

Proposition 3.3 ca μ ( , X ) is a closed set in (ca( , X ), T s ).
Proof Let m ∈ ca( , X ) and m ∈ cl T s (ca μ ( , X )). Then there is a net (m α ) in ca μ ( , X ) such that m α → m for T s , i.e., for each p ∈ P ξ and A ∈ we have Now we establish some terminology (see [14, pp. 92-93]). For p ∈ P ξ let X p = (X, p) be the associated seminormed space. Denote by ( X p , · ∼ p ) the completion of the quotient normed space X/ p −1 (0). Let p : X p → X/ p −1 (0) ⊂ X p be the canonical quotient map.
Given a vector measure m : → X with m μ, let m p : → X p be given by Then m p is a Banach space-valued measure on . We define the p-variation m p of m by where m p denotes the semivariation of m p : → X p . Note that m is ξ -bounded if and only if m p ( ) < ∞ for each ξ -continuous seminorm p on X. Moreover, we have (see [14,Lemma 7]): For a subset M of ca μ ( , X ) let Now we are ready to state a characterization of relative compactness in the space (ca μ ( , X ), T s ) in terms of the topological properties of the set K M (see [8,Theorem 7], [15,Theorem 8], [16, Theorem 2.1]). (X, ξ) be a quasicomplete lcHs. Then for a set M in ca μ ( , X ) the following statements are equivalent: This means that the family M is uniformly μ-absolutely continuous.

Theorem 3.4 Let
(iii) ⇒(iv) Assume that (iii) holds. Then M ⊂ ca μ ( , X ) ⊂ ca( , X ) and M is a uniformly ξ -countably additive set in ca( , X ). Hence by [8,Theorem 7] M is a relatively compact set in (ca( , X ), T s ). Since ca μ ( , X ) is closed in (ca( , X ), T s ), we obtain that M is a relatively compact set in (ca μ ( , X ), T s ).
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