Journal of Function Spaces

Volume 2018, Article ID 9380350, 9 pages

https://doi.org/10.1155/2018/9380350

## Integration in Orlicz-Bochner Spaces

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

Correspondence should be addressed to Marian Nowak; lp.arogz.zu.eimw@kawon.m

Received 25 January 2018; Accepted 31 March 2018; Published 14 May 2018

Academic Editor: Alberto Fiorenza

Copyright © 2018 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 be a complete -finite measure space, be a Young function, and and be Banach spaces. Let denote the Orlicz-Bochner space, and denote the finest Lebesgue topology on . We study the problem of integral representation of -continuous linear operators with respect to the representing operator-valued measures. The relationships between -continuous linear operators and the topological properties of their representing operator measures are established.

#### 1. Introduction and Preliminaries

Throughout the paper, and denote real Banach spaces and and denote their Banach duals, respectively. By and we denote the closed unit ball in and in . Let stand for the space of all bounded operators from and , equipped with the uniform operator norm .

We assume that is a complete -finite measure space. Denote by the -ring of sets with . By we denote the linear space of -equivalence classes of all strongly -measurable functions , equipped with the topology of convergence in measure on sets of finite measure.

Now we recall the basic concepts and properties of Orlicz-Bochner spaces (see [1–6] for more details).

By a* Young function* we mean here a continuous convex mapping that vanishes only at 0 and as and as . Let stand for the complementary Young function of in the sense of Young.

Let (resp., ) denote the* Orlicz-Bochner space* (resp.,* Orlicz space*) defined by a Young function ; that is,Then , equipped with the topology of the normis a Banach space. For a sequence in , if and only if for all . LetLetThen is a -closed subspace of .

Recall that a subset of is said to be* solid* whenever -a.e. and , imply . A linear topology on is said to be* locally solid* if it has a local basis at 0 consisting of solid sets (see [4]).

According to [7, Definition 2.2] and [6] we have the following definition.

*Definition 1. *A locally solid topology on is said to be a* Lebesgue topology* if for a net in , in the Banach lattice implies in .

In view of the super Dedekind completeness of one can restrict in the above definition to usual sequences in (see [7, Definition 2.2, p. 173]).

Note that, for a sequence in , in if and only if -a.e. and -a.e. for some .

For let . Then the family of all sets of the form:where is a sequence of positive numbers and is a local basis at 0 for a linear topology on (see [4, 6] for more details). Using [4, Lemma 1.1] one can show that the sets of the form are convex and solid, so is a locally convex-solid topology.

We now recall terminology and basic facts concerning the spaces of weak-measurable functions (see [8, 9]). Given a function and , let for . By we denote the linear space of the -equivalence classes of all -measurable functions . In view of the super Dedekind completeness of the set is order bounded in for each . Thus one can define the so-called* abstract norm* byOne can easy check that the following properties of hold: if and only if and , for and , if , for and .

It is known that, for , , the function defined by is measurable andMoreover, for . LetClearly . If, in particular, has the Radon-Nikodym property (i.e., is an* Asplund space*; see [10, p. 213]), then .

Let stand for the Banach dual of , equipped with the conjugate norm .

Recall that a Young function satisfies the *-condition* if for some and all . We shall say that a Young function is* completely weaker* than another (in symbols, ) if for an arbitrary there exists such that for all . Note that a Young function satisfies the -condition if and only if . If , then and it follows that .

Now we present basic properties of the topology on .

Theorem 2. *Let be a Young function. Then the following statements hold:*(i)* and if satisfies the -condition.*(ii)* is the finest Lebesgue topology on .*(iii)* is generated by the family of norms .*(iv)*, where for ,*(v)* is a closed subset of the Banach space .*(vi)*If has the Radon-Nikodym property, then the space is strongly Mackey; hence coincides with the Mackey topology .*

*Proof. *(i)–(iii) See [4, Theorems 6.1, 6.3 and 6.5].

(iv) In view of [6, Corollary 4.4 and Theorem 1.2], we get , where stands for the order continuous dual of (see [7, 8, 11] for more details). According to [8, Theorem 4.1] .

Using [11, Theorem 1.3] for we have(v) See [12, § 3, Theorem 2].

(vi) See [6, Theorem 4.5].

*Let (briefly ) denote the natural mixed topology on ; that is, is the finest linear topology that agrees with on -bounded sets in (see [5, 13, 14] for more details). Then is a locally convex-solid Hausdorff topology (see [14, Theorem 3.2]) and and have the same bounded sets. This means that is a generalized DF-space (see [15]) and its follows that is quasinormable (see [15, p. 422]). Moreover, for a sequence in , in if and only if in and (see [14, Theorem 3.1]).*

*We say that a Young function increases essentially more rapidly than another (in symbols, ) if for arbitrary , as and .*

*Theorem 3. Let be a Young function. Then the mixed topology on is generated by the family of norms .*

*Proof. *It is known that the mixed topology on is generated by the family of norms (see [16, Theorem 2.1]). Since for , by [14, (54), p. 97], the mixed topology on is generated by the family of norms .

*Since implies , in view of Theorems 2 and 3, we get*

*The problem of integral representation of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces in terms of the corresponding operator-valued measures has been the object of much study (see [5, 17–24]). In particular, Dinculeanu (see [19, § 13, Sect. 3], [20], [21, § 8, Sect. B]) studied the problem of integral representation of bounded linear operators from to a Banach space . It is known that if , and an operator measure vanishes on -null sets and has the finite -semivariation , then one can define the integral for all . Moreover, if is a bounded linear operator, then the associated operator measure has the finite -semivariation and for all (see [19, § 13, Theorem 1 p. 259], [20, Theorem 4]). The relationships of the -semivariation to the properties of operators from to were studied in [22]. Diestel [23] found the integral representation of bounded linear operators from an Orlicz-Bochner space to a Banach spaces if and a Young satisfies the -condition.*

*The present paper is a continuation of [5], where we establish integral representation of -continuous linear operators . We study the problem of integration of functions in with respect to the representing operator measures of -continuous linear operators . An integral representation theorem for -continuous linear operators is established (see Theorem 9 below). We study the relationships between -continuous operators and the properties of their representing measures .*

*2. -Semivariation of Operator Measures*

*Assume that is an additive measure such that ; that is, if .*

*Let denote the space of all -valued -simple functions on . Then if , where is a finite pairwise disjoint sequence in and . For and , we can define the integral byNote thatFor , we define a measure by the equality*

*For and , we define the integral by the equality:Then*

*Following [23], [19, § 13] one can define the -semivariation of on bywhere the supremum is taken over all finite pairwise disjoint sets in and for such that .*

*One can observe thatNote thatLet stand for the -semivariation of on ; that is,*

*The following lemma will be useful.*

*Lemma 4. Let be a Young function and be a measure with and . Then the following statements hold:(i)If , then there exists a -Cauchy sequence in such that -a.e.(ii)If is a -Cauchy sequence in , then for , is a Cauchy sequence in a Banach space and for every , is a Cauchy sequence in .(iii)If and and are -Cauchy sequence in such that -a.e. and -a.e., then for , one has and for every , one has*

*Proof. *(i) Let . Then there exists a sequence in such that -a.e. and -a.e. for all (see [21, Theorem 6, p. 4]). Using the Lebesgue dominated convergence theorem, we obtain that for all , so . Hence is a -Cauchy sequence.

(ii) Assume that is a -Cauchy sequence in . Hence for , we haveIt follows that is a Cauchy sequence in . Hence in view of (15), for , is a Cauchy sequence in .

(iii) Note that is a -Cauchy sequence and -a.e. Hence there exists such that . Note that . Hence in and it follows that there exists a subsequence of such that -a.e. Then -a.e., so and for , we getIt follows thatand hence, in view of (15) for every , we have

*Following [21, § 13, Definition 1, p. 254], in view of Lemma 4 we have the following.*

*Definition 5. *Let be a Young function and be an additive measure such that and . Then for every and , we can define the* integral* by the equalityand for , we can define the* integral* by the equalitywhere is an arbitrary -Cauchy sequence in such that -a.e.

*3. Integral Representation of Continuous Operators on Orlicz-Bochner Spaces*

*3. Integral Representation of Continuous Operators on Orlicz-Bochner Spaces*

*For a bounded linear operator let*

*Proposition 6. Let be a bounded linear operator andThen the following statements hold:(i)For and .(ii).(iii) if with .(iv) is countably additive; that is, if is a pairwise disjoint sequence in with .(v).*

*Proof. *(i) Let . Then for , we have and henceso .

(ii) This follows from (i) because if .

(iii) Assume that with . Then for . By the Lebesgue dominated convergence theorem, we obtain that for every . This means that and by (i), .

(iv) Assume that is a pairwise disjoint sequence in with . Let for . Then and . Hence by (iii) .

Statement (v) is obvious.

*Definition 7. *Let be a bounded linear operator andThen the measure will be called a* representing measure* of .

*Proposition 8. Let be a -continuous linear operator and be its representing measure. Then there exists a Young function such that and .*

*Proof. *According to Theorem 2 there exist a finite set of Young functions with for and such thatLet for . Then is a Young function with andHence

*For a linear operator and , let*

*Now we can state our main result that extends the classical results concerning the integral representation of operators on Lebesgue-Bochner spaces (see [19, § 13, Theorem 1, pp. 259–261]) to operators on Orlicz-Bochner spaces .*

*Theorem 9. Let be a -continuous linear operator and be its representing measure. Then for the following statements hold:(i) is a -continuous linear operator.(ii)For , one has and for , one has(iii)For , the measure defined by the equality is countably additive.(iv) and for , .(v).(vi)For , one has and for , one has*

*Proof. *(i) Assume that is a net in such that in . Since is a locally solid topology on , we get in . Hence(ii) In view of Proposition 8 there exists a Young function such that and . Then . Let . Then there exists a sequence in such that -a.e. and -a.e. for all (see [21, Theorem 6, p. 4]). Then in because is a Lebesgue topology. Hence . In view of Lemma 4 we can define the integral by the equalitySince and by (i), is -continuous, we getHenceand for , we have (iii) Let and be a sequence in such that . Then for , and hence -a.e. and -a.e. Hence in because is a Lebesgue topology, and by (i) we get (iv) Note that . To show that , assume that . Choose a sequence in such that -a.e. and -a.e. for all . Since is a Lebesgue topology, we have in and hence . Note that .

Let be given. Choose such that . ThenIt follows that , so . Hence for , we easily get (v) Using (iv) we have (vi) This follows from (ii) and (iv).

*For a sequence in , we will write if and for every .*

*Definition 10. *A measure with and is said to be *-semivariationally **-continuous* if whenever , .

*Using a standard argument we can show the following.*

*Proposition 11. Let be an additive measure such that and . Then the following statements are equivalent:(i) is -semivariationally -continuous.(ii)The following two conditions hold simultaneously:(a)For every there exists such that whenever , .(b)For every there exists such that .*

*The following theorem characterizes -semivariationally -continuous representing measures.*

*Theorem 12. Let be a -continuous linear operator and be its representing measure. Then the following statements are equivalent:(i) is -semivariationally -continuous.(ii) is -continuous.(iii) if in and .(iv) if , .*

*Proof. *(i)* ⇔* (ii)

*(iii) See [5, Corollary 2.8 and Proposition 1.1].*

*⇔*(i)

*(iv) This follows from Theorem 9.*

*⇔**Now assume that is a completely regular Hausdorff space. Let denote the -algebra of Baire sets in , which is the -algebra generated by the class of all zero sets of bounded continuous positive functions on . By we denote the family of all cozero (=positive) in (see [25, p. 108]).*

*Let be a countably additive measure. Then is zero-set regular; that is, for every and there exists with such that (see [25, p. 118]). It follows that for every and there exist , such that .*

*We can assume that to be complete (if necessary we can take the completion of the measure space ).*

*Proposition 13. Assume that is a completely regular Hausdorff space and is a complete finite measure space. Let be a -continuous linear operator and be its representing measure. Then the following statements are equivalent:(i) is -semivariationally -continuous.(ii)For every sequence in such that and there exists a sequence in with such that .(iii)For every sequence in such that and there exists a sequence in with such that*

*Proof. *(i) ⇒ (ii) Assume that (i) holds and is a sequence in such that and . Then there exists a sequence in such that and for .

Let be given. Then in view of Proposition 11 there exists such that if with . Choose such that for . Then for . Since , we can choose such that for . Then for , we getthat is, (ii) holds.

(ii) ⇒ (iii) Assume that (ii) holds and is a sequence in such that and . Then there exists a sequence in with such that . Note that, for with for , by Theorem 9 we haveIt follows that (iii) holds.

(iii) ⇒ (i) Assume that (iii) holds and with . Then there exists a sequence in with such thatAssume on the contrary that (i) fails to hold. Then without loss of generality we can assume thatChoose such thatIn view of (54) there exists a pairwise disjoint set in , for and such that andLet . Then and . Then by (55) we get .

On the other hand, in view of (56) we have . This contradiction establishes that (i) holds.

*Corollary 14. Assume that is a completely regular Hausdorff space and is complete finite measure space. Let be a -continuous linear operator and be its representing measure. Then is regular; that is, for every and there exist and with such that .*

*Proof. *In view of Theorem 12 is -semivariationally -continuous. Let