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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 343685, 8 pages
http://dx.doi.org/10.1155/2013/343685
Research Article

Topological Properties of the Complex Vector Lattice of Bounded Measurable Functions

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Szafrana 4A, 65-516 Zielona Góra, Poland

Received 21 May 2013; Accepted 12 August 2013

Academic Editor: Stanislav Hencl

Copyright © 2013 Marian Nowak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Σ be a σ-algebra of subsets of a nonempty set Ω. Let be the complex vector lattice of bounded Σ-measurable complex-valued functions on Ω and let be the Banach space of all bounded countably additive complex-valued measures on Ω. We study locally solid topologies on . In particular, it is shown that the Mackey topology is the finest locally convex-solid σ-Lebesgue topology on .

1. Introduction and Terminology

For terminology concerning complex vector lattices we refer the reader to [1, Chapter 12, Section  91], [2, Chapter 16], [3]. We denote by and the weak topology and the Mackey topology with respect to a dual pair . For a topological vector space by or , we denote its topological dual.

We assume that is a -algebra of subsets of a non-empty set . Let denote the Banach space of all bounded -measurable functions , provided with the uniform norm . Then, with the positive cone for all and a valuation is a complex vector lattice (in the sense of [2, Section  16-2.2]). By we denote the Dedekind -complete Banach lattice of all bounded -measurable functions , with the natural ordering if for all (see [4, Chapter 13.1]). Then, , that is, is a complexification of and has the dominated decomposition property, that is, if , where , , then there exist such that and , (see [1, Theorem 91.3], [3]). This means that is a breakable complex vector lattice (see [2, Section 16-3]).

Denote by the Banach space of all bounded finitely additive measures with the norm , where denotes the variation of on . It is well known that the Banach dual of can be identified with through the integration mapping , where for all and (see [5, Chapter 1, Theorem 13, p.6]. Let stand for the Banach space of all countably additive measures .

It turns out that the basic notions and results from the theory of vector lattices (in particular, locally solid topologies) can be carried over to the complex vector lattices and .

A subset of is said to be solid if it follows from and , that . Every subset of is included in the smallest (with respect to the inclusion) solid set, called the solid hull of and denoted by . Clearly,

A linear Hausdorff topology on is said to be locally solid if it has a local base at consisting of solid sets in .

In Section 2, we study locally solid topologies on . In Section 3, we develop the duality theory of locally convex-solid topologies on . In Section 4, it is shown that the natural Mackey topology is the finest locally convex-solid -Lebesgue topology on (see Theorem 16 below).

2. Locally Solid Topologies on

Using the dominated decomposition property for we can easily obtain the following result (see [6, Theorem 1.11]).

Proposition 1. The convex hull of a solid subset of is solid.

Proposition 2. Let be a locally solid topology on . Then, the -closure of a solid subset of is solid.

Proof. Let be a local base at for consisting of solid sets. Then . Assume that , where , , and let . Then, , where and . Since (by the dominated decomposition property for ), there exist such that and for . Hence, and because both sets and are solid. Thus, for every , so . This means that is solid, as desired.

Following the notion of a modulus of an operator between vector lattices (see [7, Definition 1.8 and Theorem 1.10]), for a linear functional let us put for all .

Proposition 3. For a linear functional on the following statements are equivalent:(i) .(ii) for  all .

Proof. (i) (ii) By way of contradiction, assume that there exists such that for some . Hence, there exists a sequence in such that and for . Since , we get , which is in contradiction with .
(ii) (i) Assume that there exists a linear functional on such that for all and . Then, there exists a sequence in such that and for all . Since and the space is complete, there exists such that . Then, , and; hence, for all , which is impossible.
Thus, the proof is complete.

For we will write whenever for all . A subset of is said to be solid if , where , implies that . A linear subspace of is said to be an ideal if is a solid set in .

Note that, if , then .

Proposition 4. Let be a locally solid topology on . Then, is an ideal of .

Proof. To show that , by the way of contradiction, assume that for some we have , so in view of Proposition 3, we obtain that for some . Hence, there exists a sequence in such that and for all . Since for and is locally solid, we get for . Hence, , which is in contradiction with for all .
To see that is an ideal of , assume that , where and . Let for and be given. Then, there exists a net in such that for each and . Clearly, for because is locally solid, so . Since , we get , so , as desired.

3. Duality of the Space

A locally convex Hausdorff topology on is said to be locally convex-solid if it has a local base at consisting of convex and solid sets in . In view of Propositions 1 and 2, we see that for a locally convex-solid topology on the collection of all -closed convex and solid -neighborhoods of forms a local base at for .

In this section, we develop the duality theory of locally convex-solid topologies on . We start with the following lemma.

Lemma 5. Let . Then, for each we have In particular, we have for

Proof. Let and with . Then, we have so . Now we shall show that . Indeed, let be given. Then, there exists a -simple function such that , and Let . Then for each one can choose a finite -partition of and with , for such that Let . Then and we have Hence, , and it follows that .

Since is additive, can be uniquely extended to a positive linear functional (denoted by again) (see [2, Theorem 16-4.7]). Then, for we have Hence, .

Proposition 6. For the following statements are equivalent: (i) .(ii) .

Proof. (i) (ii) Assume that . Then, by Lemma 5, for , , that is, .
(ii) (i) Assume that . Then, for we have .

Now we consider the concept of solidness in the space . A subset of is said to be solid if , where , implies that .

For a subset of let Then, is the solid hull of , that is, is the smallest solid set in that contains .

A linear subspace of is called an ideal in if is solid. Note that is an ideal of . In view of Proposition 6, we see that a subset of is solid if and only if the set is solid in .

Corollary 7. Let . Then, for we have Moreover, for a subset of and each we have

Proof. In view of Proposition 6 and [2, Theorem 16-5.6] we get Next, we shall show that for each , Indeed, let be given. Then, for and by there exists with and such that . Since , we get , and it follows that (12) holds.
It is enough to show that for each we have Indeed, let . Then, with , and . Hence, for some , . Hence, for we get It follows that (13) holds, as desired.

Let be an ideal of separating the points of . Then, the pair , under its natural duality will be referred to as a solid dual system (see [7, Definition 11.7]). For subsets of and of , let us put

Proposition 8. Let be a solid dual system.(i)If is a solid subset of , then is a solid subset of .(ii)If is a solid subset of , then is a solid subset of .

Proof. (i) Let with and . Assume that and let with . Then, because is solid, so . Hence (see Lemma 5). Thus , that is, . This means that is a solid subset of .
(ii) Let , where and . Since is a solid set in , by Corollary 7 for , we have Then, for every with we have , so . It follows that that is, .

Observe that for each ,  by Corollary 7 the functional defined by is a solid seminorm on , that is, if .

Let be an ideal of separating the points of . The absolute weak topology generated by on is the locally convex-solid topology on generated by the family of solid seminorms . Note that is the topology of uniform convergence on all sets , where .

Proposition 9. For each , the set is convex, solid, and -compact in .

Proof. Clearly, is a convex and solid subset of . We shall show that is -closed subset of . To this end, assume that is a net in such that for all , where . Then Hence, for each , we get for all , and by Lemma 5, that is, . Thus, , as desired.
In view of the Banach-Alaoglu theorem, is a relatively -compact subset of . Since is -closed in , is -compact, and it follows that is -compact.

We define the absolute Mackey topology   as the topology of uniform convergence on all convex, solid and -compact subsets of . By Proposition 8   is a locally convex-solid topology on . Thus, in view of Proposition 9 we have

As a consequence, we obtain the following Mackey-Arens type theorem for locally convex-solid topologies on .

Proposition 10. Let be a locally convex-solid topology on . Then, .

Proof. Note that the family of all convex, solid and -closed -neighborhoods of forms a local base at for . Then, is the topology of uniform convergence on the family . By Proposition 8 and the Banach-Alaoglu theorem, for each , is convex, solid, and -compact in . It follows that .
To show that , assume that in for . Then, for because is locally solid. Let be such that . Then, because is an ideal of (see Proposition 4). Hence, , and it follows that for , as desired.

4. The Mackey Topology

In this section we study the Mackey topology on . It is known that is a generalized DF-space, that is, is the finest locally convex topology agreeing with itself on norm-bounded sets in (see [8, Corollary 11.8], [9, Section  4], [10, Theorem 2]).

Let stand for the Dedekind complete Banach lattice of all countably additive measures . Then, is a band of . Moreover, the -order continuous dual of can be identified with through the integration mapping , where for all (see [4, Theorem 13.5]).

Recall that a sequence in is said to be order convergent to (in symbols,   if there exists a sequence in such that (in the vector lattice sense; see [6, Definition  1.12]).

Definition 11. A linear functional on is said to be -order continuous if for each sequence in , in implies .

Proposition 12. For the following statements are equivalent:(i) is -order continuous. (ii) for each uniformly bounded sequence in such that for all . (iii) for each sequence in such that for all . (iv) .

Proof. (i) (ii) Assume that is -order continuous and let for all and . Let for , . Then, and for all and . It follows that in , and hence, .
(ii) (iii) It is obvious.
(iii) (iv) Assume that (iii) holds, and let , . Then, for all . Hence, .
(iv) (i) Assume that . Let be a sequence in such that in , that is, for some sequence in . Then Note that for -almost all . Indeed, let for each . Then, , so . If , then there exists such that satisfies . But then for each , contradicting in . Hence, by Lebesgue Dominated Convergence theorem, we get , so .

By we will denote the space of all -order continuous functionals on . For let denote the evaluation functional on , that is, for all . Then and it follows that separates the points of .

From Proposition 12 and Lemma 5 if follows that for , if and only if . Since is an ideal of , we see that is an ideal of . For a subset of let

Proposition 13. For a subset of the following statements are equivalent:(i) is relatively -compact. (ii) is relatively -countably compact. (iii) and is uniformly countably additive. (iv) and is uniformly countably additive. (v) is relatively weakly compact. (vi) is relatively weakly compact.

Proof. (i) (ii) It is obvious.
(ii) (iii) Assume that is relatively -countably compact. It follows that is -bounded, and since , by the uniform boundedness theorem, , that is, .
Assume, on the contrary, that is not uniformly countably additive. Then there exist a sequence in , a sequence in and such that and for all . Let . Putting , we see that and for all . Hence, is a relatively -countably compact subset of (= the space of all -absolutely continuous measures in ). In view of the Radon-Nikodym theorem, for each there exists uniquely such that for each . It follows that for every we have . Hence, the linear mapping is -continuous. Thus, the set is relatively -countably compact in . Hence, by the Eberlein-Smulian theorem is relatively -compact in . According to the Dunford-Pettis theorem (see [5, Chapter 3, Theorem  15]), the set in is uniformly integrable. Since , we have that , and hence , contradicting for .
(iii) (iv) Assume that and is uniformly countably additive. Then, is uniformly countably additive (see [5, Chapter 1, Proposition  17]). Note that for , . It follows that and is uniformly countably additive. Moreover, one can easily observe that for , and . It follows that is uniformly bounded and uniformly countably additive.
(iv) (v) It is well known (see [11, Chapter 7, Theorem  13]).
(v) (vi) (i) It is obvious.

Recall that a locally convex space is said to be strongly Mackey if every relatively -countably compact set in is -equicontinuous.

Corollary 14. The space is strongly Mackey.

Proof. Let be a relatively -countably compact set in . Then, by Proposition 13, the set is absolutely convex and -compact. Hence, is a -neighborhood of . It follows that is -equicontinuous.

Now we distinguish three natural classes of linear topologies on .

Definition 15. A locally solid topology on is said to be(i) -Lebesgue if for whenever is a sequence in such that in (in the vector lattice sense).(ii)a -smooth topology if for whenever is uniformly bounded sequence in such that for all .(iii)a -Dini topology if for whenever is a sequence in such that for all .

Note that every -Lebesgue topology on is -smooth, and every -smooth topology on is -Dini.

Now, we characterize the Mackey topology on .

Theorem 16. (i) is a locally convex-solid topology.
(ii) is the finest locally convex -Lebesgue topology on .
(iii) is the finest locally convex -smooth topology on .
(iv) is the finest locally convex -Dini topology on .

Proof. (i) Let be an absolutely convex and closed neighborhood of for . Then, the polar of (with respect to the solid dual system is -compact. In view of Proposition 13   is relatively -compact and hence (the closure is taken in is absolutely convex and -compact. Hence, It follows that is a convex and solid neighborhood of for (see Proposition 8). This means that is a locally convex-solid topology on .
(ii) Now we shall show that is a -Lebesgue topology. Assume that is a sequence in such that in the vector lattice . Let be a relatively -compact set in . Then is uniformly bounded and uniformly countably additive (see Proposition 13), and it follows that is uniformly countably additive. Hence, by [10, Theorem 3] is -equicontinuous. Hence, the polar of (with respect to the dual pair is a -neighborhood of in . Let be given. Since is a -Lebesgue topology on (see [6, Example 18, page 178], [6, Corollary 6.29 and Theorem 3.20]), there exists such that for . Hence for we get It follows that for , and this means that is a -Lebesgue topology on .
Now assume that is a locally convex-solid -Lebesgue topology on . Then , and hence, .
(iii) From (ii) it follows that is a -smooth topology.
Now let be a locally convex -smooth topology on . Then by Proposition 12 we get . Hence, .
(iv) Clearly is a -Dini topology on . Assume that is a locally convex -Dini topology on . In view of Proposition 12, and it follows that .

A characterization of relative weak compactness in the Banach space in terms of uniform boundedness and uniform countable additivity of measures belongs to the classical results in measure theory (see [11, Chapter 7, Theorem 13], [8, Theorem 11.6], [12, Theorem 1.1]). Now, using the properties of the Mackey topology we are ready to add Grothendieck type conditions to this characterization.

Corollary 17. For a subset of the following statements are equivalent:(i) is a relatively weakly compact. (ii) and uniformly for whenever is a uniformly bounded sequence in such that for all .(iii) and uniformly for whenever is a sequence in such that for all .

Proof. (i) (ii) Assume that is relatively weakly compact. Then, in view of Corollary 14   is bounded and -equicontinuous. Let be a uniformly bounded sequence in such that for all . Since is a -smooth topology on (see Theorem 16), we obtain that for . Now let be given. Then, there exists a -neighborhood of such that for all . Hence, there exists such that for , and it follows that for , as desired.
(ii) (iii) It is obvious.
(iii) (i) Assume that (iv) holds and let , . Then, for all . Hence , that is, is uniformly countably additive. Hence, is relatively weakly compact (see [11, Chapter 7, Theorem 13]).

Remark 18. One can observe that Corollary 17 is closely related to a Grothendieck’s characterization of relative weak compactness in the space of bounded complex Radon measures on a locally compact space (see [13, Theorem 2]).

Now we define a class of linear operators on .

Definition 19. Let be a locally convex Hausdorff space (briefly, lcHs). A linear operator is said to be -order continuous if for each sequence in in implies for .

For terminology and basic results concerning the integration with respect to vector measures, we refer the reader to [1417].

Let be a quasicomplete lcHs. Let be a -bounded measure (i.e., the range of is -bounded in ). Given , let be a sequence of -simple scalar functions that converges uniformly to on . Following [14, Definition 1] we say that is -integrable and define The is well defined (see [14, Lemma 5]) and the map given by is -continuous and linear, and for each (see [14, Lemma 5]). Conversely, let be a -continuous linear operator, and let for . Then, is a -bounded vector measure, called the representing measure of and for (see [14, Definition 2]).

Recall that a measure is said to be -countably additive if for whenever , .

The following characterization of -order continuous operators from into a quasicomplete lcHs displays the close connection between the Mackey topology on and -valued -countably additive measures.

Proposition 20. Assume that is a quasicomplete lcHs. Then for a -bounded measure the following statements are equivalent:(i) is -continuous. (ii) is -continuous. (iii) is sequentially -continuous. (iv) is -order continuous. (v) for whenever is a uniformly bounded sequence in such that for all .(vi) for for each sequence in such that for all .(vii) is -countably additive.

Proof. (i) (ii) (iii) It is obvious.
(iii) (iv) Assume that is sequentially -continuous, and let be a sequence in such that in . Then, by Theorem 16, for . Hence, for , that is, is -order continuous.
(iv) (v) Assume that is -order continuous, and assume that is a uniformly bounded sequence in such that for all . Then in (see the proof of Proposition 12). Hence, for , as desired.
(v) (vi) It is obvious.
(vi) (vii) Assume that (vi) holds, and let , . Then, for all , and it follows that for . This means that is -countably additive.
(vii) (i) Assume that is -countably additive. Then for each , and in view of (28) and Proposition 12 we get for each . Hence, is -continuous (see [7, Theorem  9.26]). It follows that is -continuous (see [7, Example 11, page 149]).

Acknowledgments

The author wishes to thank the referee for useful suggestions that have improved the paper.

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