#### 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 [14–17].

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.