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Vector Measures and Mackey Topologies View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Available online at www.sciencedirect.com Indagationes Mathematicae 23 (2012) 113–122 www.elsevier.com/locate/indag Vector measures and Mackey topologies Marian Nowak Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora,´ ul. Szafrana 4A, 65–516 Zielona Gora,´ Poland Received 15 October 2011; accepted 6 November 2011 Communicated by B. de Pagter Abstract Let Σ be a σ -algebra of subsets of a non-empty set Ω. Let B.Σ/ be the Banach lattice of all bounded Σ-measurable real-valued functions defined on Ω, equipped with the natural Mackey topology τ.B.Σ/; ca.Σ//. We study (τ.B.Σ/; ca.Σ//; ξ/-continuous linear operators from B.Σ/ to a quasicomplete locally convex space .E; ξ/. A generalized Nikodym convergence theorem and a Vitali–Hahn–Saks type theorem for operators on B.Σ/ are obtained. It is shown that the space .B.Σ/; τ.B.Σ/; ca.Σ/// has the strict Dunford–Pettis property. Moreover, a Yosida–Hewitt type decomposition for weakly compact operators on B.Σ/ is given. ⃝c 2011 Royal Netherlands Academy of Arts and Sciences. Published by Elsevier B.V. All rights reserved. Keywords: Bounded measurable functions; σ-smooth operators; Mackey topologies; Strongly Mackey space; Weakly compact operators; Countably additive measures; Strict Dunford–Pettis property 1. Introduction and terminology Properties of bounded linear operators from the space B.Σ/ to a Banach space E can be expressed in terms of the properties of their representing vector measures (see [6, Theorem 2.2], [7, Theorem 1, p. 148], [11, Corollary 12], [16, Theorem 10], [18, Theorem 2.1]). In this paper we study linear operators from B.Σ/ to a quasicomplete locally convex space .E; ξ/. In particular, we obtain a Vitali–Hahn–Saks type theorem and a Nikodym convergence type theorem for (τ.B.Σ/; ca.Σ//; ξ/-continuous linear operators T V B.Σ/ ! E. It is shown that the space E-mail address: [email protected]. 0019-3577/$ - see front matter ⃝c 2011 Royal Netherlands Academy of Arts and Sciences. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.indag.2011.11.001 114 M. Nowak / Indagationes Mathematicae 23 (2012) 113–122 .B.Σ/; τ.B.Σ/; ca.Σ/// has the strict Dunford–Pettis property. Moreover, a Yosida–Hewitt type decomposition for weakly compact operators on B.Σ/ is given. For terminology concerning vector lattices we refer the reader to [2,3,1]. We denote by σ .L; K /; τ.L; K / and β.L; K / the weak topology, the Mackey topology and the strong topology on L with respect to the dual pair hL; K i. We assume that .E; ξ/ is a locally convex Hausdorff 0 0 space (for short, lcHs). By .E; ξ/ or Eξ we denote the topological dual of .E; ξ/. Recall that 0 0 .E; ξ/ is a strongly Mackey space if every relatively countably σ .Eξ ; E/-compact subset of Eξ 00 00 D 0 0 0 is ξ-equicontinuous. By Eξ we denote the bidual of .E; ξ/, i.e., Eξ .Eξ ; β.Eξ ; E// . Let Σ be a σ-algebra of subsets of a non-empty set Ω. By ca.Σ; E/ we denote the space of all ξ-countably additive vector measures m V Ω ! E, and we will write ca.Σ/ if E D R. By S.Σ/ we denote the space of all real-valued Σ-simple functions defined on Ω. Then S.Σ/ can be endowed with the (locally convex) universal measure topology τ of Graves [10], that is, τ is the coarsest locally convex topology on S.Σ/ such that the integration map Tm V S.Σ/ 3 7! R 2 2 s Ω s dm E is continuous for every locally convex space .E; ξ/ and every m ca.Σ; E/ (see [10, p. 5]). Let .L.Σ/; τ/O stand for the completion of .S.Σ/; τ/. It is known that both .S.Σ/; τ/ and .L.Σ/; τ/O are strongly Mackey spaces (see [10, Corollaries 11.7 and 11.8]). It follows that τ D τ.S.Σ/; ca.Σ// and τO D τ.L.Σ/; ca.Σ// (see [10,11]). Moreover, if .E; ξ/ is complete in its Mackey topology, then for each m 2 ca.Σ; E/, the integration map Tm can be uniquely extended to a .τ;O ξ/-continuous map Tem V L.Σ/ ! E (see [11]). Let B.Σ/ denote the Dedekind σ-complete Banach lattice of all bounded Σ-measurable functions u V Ω ! R, provided with the uniform norm k · k. Then B.Σ/ is the k · k-closure of S.Σ/, so S.Σ/ ⊂ B.Σ/ ⊂ L.Σ/ and the restriction τO from L.Σ/ to B.Σ/ coincides with the Mackey topology τ.B.Σ/; ca.Σ// (see [11, p. 199]). Moreover, .B.Σ/; τOjB.Σ// is a strongly Mackey space (see [10, Corollary 11.8]). Note that the topology γ1 on B.Σ/ studied by Khurana [12], coincides with the topology τOjB.Σ/.Dτ.B.Σ/; ca.Σ/// (see [12, Theorem 2, Corollary 6]). Denote by ba.Σ/ the Banach lattice of all bounded finitely additive measures ν V Σ ! R with the norm kνk D jνj.Ω/, where jνj.A/ denotes the variation of ν on A 2 Σ. It is well known that the Banach dual B.Σ/∗ of B.Σ/ can be identified with ba.Σ/ through the mapping ∗ ba.Σ/ 3 ν 7! Φν 2 B.Σ/ , where Z Φν.u/ D u dν for u 2 B.Σ/: Ω ∗ Then kΦνk D jνj.Ω/ (see [1, Theorem 13.4]). The σ-order continuous dual B.Σ/c of B.Σ/ ∗ ∗ is a band of B.Σ/ (separating the points of B.Σ/) and B.Σ/c can be identified with ca.Σ/ (see [1, Theorem 13.5]). Hence 0 0 ∗ .B.Σ/; τOjB.Σ// D .B.Σ/; τ.B.Σ/; ca.Σ/// D B.Σ/c : Moreover, it is well known that τ.B.Σ/; ca.Σ// is a locally solid σ-Lebesgue topology on B.Σ/ (see [3, Ex. 18, p. 178], [12, Theorem 3]). 2. σ-smooth operators on B.Σ/ In this section we study (τ.B.Σ/; ca.Σ//; ξ/-continuous linear operators from B.Σ/ to a locally convex Hausdorff space .E; ξ/. Recall that a sequence .un/ in B.Σ/ is order convergent to .o/ u 2 B.Σ/ (in symbols, un −! u) if there is a sequence .vn/ in B.Σ/ such that jun −uj ≤ vn # 0 in B.Σ/ (see [3]). M. Nowak / Indagationes Mathematicae 23 (2012) 113–122 115 Definition 2.1. A linear operator T V B.Σ/ ! E is said to be σ-smooth if T .un/ ! 0 in ξ .o/ whenever un −! 0 in B.Σ/. Proposition 2.1. For a linear operator T V B.Σ/ ! E the following statements are equivalent: 0 ◦ 2 ∗ 0 2 0 (i) e T B.Σ/c for each e Eξ . 0 (ii) T is (σ .B.Σ/; ca.Σ//; σ .E; Eξ //-continuous. (iii) T is (τ.B.Σ/; ca.Σ//; ξ/-continuous. (iv) T is (τ.B.Σ/; ca.Σ//; ξ/-sequentially continuous. (v) T is σ-smooth. Proof. (i) () (ii) See [2, Theorem 9.26]. D) 0 (ii) (iii) Assume that T is (σ .B.Σ/; ca.Σ//; σ .E; Eξ //-continuous. Then T is 0 (τ.B.Σ/; ca.Σ//; τ.E; Eξ //-continuous (see [2, Ex. 11, p. 149]). It follows that T is ⊂ 0 (τ.B.Σ/; ca.Σ//; ξ/-continuous because ξ τ.E; Eξ /. (iii) D) (iv) It is obvious. .o/ (iv) D) (v) Assume that T is (τ.B.Σ/; ca.Σ//; ξ/-sequentially continuous, and let un −! 0 in B.Σ/. Then un ! 0 for τ.B.Σ/; ca.Σ// because τ.B.Σ/; ca.Σ// is a σ-Lebesgue topology. Hence T .un/ ! 0 for ξ, i.e., T is σ -smooth. (v) D) (i) It is obvious. Note that every σ-smooth operator T V B.Σ/ ! E is .k · k; ξ/-continuous because τ.B.Σ/; ca.Σ// ⊂ T∥·∥. Let Lτ,ξ .B.Σ/; E/ stand for the space of all (τ.B.Σ/; ca.Σ//; ξ/-continuous linear operators from B.Σ/ to E, equipped with the topology Ts of simple convergence. Then Tα ! T for Ts in Lτ,ξ .B.Σ/; E/ if and only if Tα.u/ ! T .u/ in ξ for all u 2 B.Σ/. The following result will be of importance (see [16, Theorem 2]). 0 Theorem 2.2. Let K be a Ts-compact subset of Lτ,ξ .B.Σ/; E/. If C is a σ .Eξ ; E/-closed and 0 f 0 ◦ V 2 0 2 g ∗ ξ-equicontinuous subset of Eξ , then e T T K; e C is a σ .B.Σ/c ; B.Σ//-compact ∗ subset of B.Σ/c . Now using Theorem 2.2 and the property that .B.Σ/; τ.B.Σ/; ca.Σ/// is a strongly Mackey space, we are ready to prove our main result. Theorem 2.3. Let K be a subset of Lτ,ξ .B.Σ/; E/. Then the following statements are equivalent: (i) K is relatively Ts-compact. (ii) K is (τ.B.Σ/; ca.Σ//; ξ/-equicontinuous and for each u 2 B.Σ/, the set fT .u/ V T 2 Kg is relatively ξ-compact in E.
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