Journal of Function Spaces
Volume 2015 (2015), Article ID 260238, 8 pages
http://dx.doi.org/10.1155/2015/260238

Research Article

New Set-Valued Integral in a Banach Space

Cai-Li Zhou¹ and Fu-Gui Shi²

¹College of Mathematics and Information Science, Hebei University, Baoding 071002, China
²School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Received 26 October 2014; Accepted 27 February 2015

Academic Editor: Wilfredo Urbina

Copyright © 2015 Cai-Li Zhou and Fu-Gui Shi. 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.

We introduce and study a new set-valued integral of scalar-valued functions with respect to a set-valued measure in a Banach space. We investigate some properties and convergence theorems for this kind of integral.

1. Introduction

The need for set-valued measures first arose in mathematical economics when Vind [1] studied equilibrium theory for exchange economies with production. Since then the subject of set-valued measures has received a lot of attention and, as it turned out, developed to be the set-valued analogue of the classical theory of vector measures (see, e.g., [2–9]). Accordingly, various developments in mathematical economics and optimal control have led to the study of integrals with respect to set-valued measures and many papers dealt with the basic theory of these integrals. Among them, as an extension of the integral of scalar-valued functions with respect to vector measures, Papageorgiou [5] first introduced the integral of scalar-valued functions with respect to a set-valued measure by the bilinear integral of Dinculeanu [10]. Since then, Kandilakis [11] introduced an integral of bounded real-valued functions with respect to a set-valued measure using the set of Kluvánek-Knowles type integrals [12]. Wu et al. [13] introduced the set-valued Bartle integral which is the set of Bartle-Dunford-Schwartz type integrals [14]. The set-valued Bartle integral has also been deeply considered by Zhang et al. [9]. Precupanu and Croitoru [15, 16] introduced a multivalued Gould type integral for a real-valued function with respect to a set-valued measure. Using [15, 16], Gavriluţ [17, 18] introduced a Gould type integral of a real-valued function with respect to a multisubmeasure. Recently, Precupanu and Satco [19] introduced an Aumann-Gould integral in which the selectors are Gould integrals of real-valued functions with respect to vector-valued measures [20]. Precupanu et al. [21] introduced a set-valued Gould type integral of a real function with respect to a set-valued fuzzy measure.

In the present paper, we introduce and study a new set-valued integral of scalar-valued functions with respect to a set-valued measure. Unlike various ways of defining such an integral (e.g., Papageorgiou way, Kandilakis way, and Wu, Zhang, and Wang way), our integral does not depend on measure selections of the given set-valued measure and is a Pettis type set-valued integral of scalar-valued functions with respect to a set-valued measure. The result is an extension of the Kluvánek-Lewis type integral [22–24], which is a Pettis type weak integral of scalar-valued functions with respect to a vector-valued measure. We investigate some properties and establish convergence theorems for this kind of integral. Finally, we discuss the relationship between this new integral and the set-valued Bartle integral.

The paper is structured as follows. In Section 2, we state some basic concepts and preliminary results which will be used in the sequel. In Section 3, we first introduce the new integral with some natural properties. And then, we prove Vitali type convergence theorem and dominated convergence theorem for this kind of integral. Finally, it is shown that, under some natural assumptions on the Banach space and on the set-valued measure, our integral is deduced to the set-valued Bartle integral, which shows that our integral is also an extension of the set-valued Bartle integral.

2. Preliminaries

Throughout this paper, let be a complete finite measure space where is a nonempty set, is a -algebra of subsets of , and is a measure. Let be a real separable Banach space with its dual space . Let

For , the Hausdorff metric of and is defined by

where . For , the number is defined by . Note that is a complete metric space.

Throughout this work we will denote by the support function of the set defined by

The support function satisfies the properties and for all ,   and . In particular,

whenever are two convex sets.

Definition 1 (see [6]). Let be a measurable space. A set-valued set function is said to be a set-valued measure if it satisfies the following two conditions:

(1);(2)if are in , with for , then

where .

In particular, is a set-valued measure if and only if is a scalar-valued measure for all (cf. [6]).

A vector-valued measure satisfying for all is said to be a selection of the set-valued measure . We denote the set of all selections of by . For a set-valued measure , we have the notion of total variation defined by for , where the supremum is taken over all finite measurable partitions of . It is easy to check that is a positive measure. If , then we say that is of bounded variation. In this case the sums in the definition of are absolutely convergent. We call that is -continuous, where is a single-valued vector measure; if, for any , , then .

A Banach space has the Radon-Nikodým property (RNP) if, for each finite measure space and each -continuous -valued measure of bounded variation, there exists a Bochner integrable function such that for all . A measurable set-valued function is said to be a Radon-Nikodým derivative of the set-valued measure with respect to if for all and we write .

Definition 2 (see [24]). Let be a vector-valued measure and let be -measurable. is said to be Kluvánek-Lewis integrable with respect to (for short, (KL) -integrable) if

(1) is -integrable for each ;(2)for each , there exists a vector such that

for each . In that case, we define for each .

Note that by the Hahn-Banach theorem we know that the integral is well defined and unique.

3. Main Results

In the sequel, let be the space of all functions which are -measurable and -integrable. We first introduce the new integral as follows.

Definition 3. Let be a set-valued measure and an element of . is said to be Kluvánek-Lewis integrable with respect to (for short, (KL) -integrable) if

(1) is -integrable for each ;(2)for each , there exists a such that

for each . In the case, we write

for each and call it set-valued Kluvánek-Lewis integral of with respect to on .

Remark 4. The set-valued Kluvánek-Lewis integral is a generalization of the Kluvánek-Lewis integral. Note that when the set-valued measure is degenerated into a vector-valued measure , the set is reduced to a vector in and the equality turns into

for each .

Example 5. Let be a set-valued measure. If is a nonnegative simple function, then is (KL) -integrable and

for each .

Proof. Obviously, is -integrable for each and

for each . On the other hand, by the properties of support functions, we have

for each . It follows from the above two equalities that

for each . Since , we have . Thus, is (KL) -integrable and

This completes the proof.

In what follows, some properties of the set-valued Kluvánek-Lewis integral will be given.

Theorem 6. Let be a set-valued measure, (KL) -integrable, and . Then

(1) is (KL) -integrable and

 for each ;(2) is (KL) -integrable and

 for each .

Proof. (1) Since and are (KL) -integrable, and are -integrable for each and there exist

in such that

for each and . Obviously, is -integrable and, by properties of support functions, we have

for each and . Since , we have . It follows that is (KL) -integrable and

for each .
(2) In the same manner as in the proof of (1), we can show that is -integrable and

for each , , and . Since , we have . Thus is (KL) -integrable and

for each . This completes the proof.

One of the most important properties of the Kluvánek-Lewis integral is that the indefinite integral of an integrable function is a vector measure. We would like to extend the result to set-valued Kluvánek-Lewis integrals as follows.

Theorem 7. Let be a nonnegative finite measure space, a -continuous set-valued measure, and (KL) -integrable. Then defined by

is a -continuous set-valued measure.

Proof. Note that is a set-valued measure if and only if is a scalar-valued measure for each . Thus, to end the proof, we show that is a scalar-valued measure for each . Obviously,

for each . In the following, we prove countable additivity of . For each and , we have

Since is -integrable, it is known from standard scalar-valued measure theory that is countably additive. Thus, if is a sequence of pairwise disjoint elements of , we have

that is, is countably additive. So, we conclude that is a scalar-valued measure for each . It follows that is a set-valued measure.
By Theorem 6.3.2 [9], is a -continuous set-valued measure if and only if is a -continuous scalar-valued measure for each . It follows that is -continuous for each . Again by Theorem 6.3.2 [9], is a -continuous set-valued measure. This completes the proof.

Theorem 8. Suppose that has the RNP, is separable, is a -continuous set-valued measure of bounded variation, and is (KL) -integrable. Then there exists an integrably bounded set-valued function such that

Proof. Since has the RNP, is separable and is a -continuous set-valued measure of bounded variation, by Theorem 6.4.4 [9], there exists an integrably bounded set-valued function which is a Radon-Nikodým derivative of with respect to . Then we have

for each . It follows that

for each and each . Note that is a -continuous set-valued measure if and only if is a -continuous scalar-valued measure for each , which, together with the above equality, implies that is a Radon-Nikodým derivative of with respect to for each . Hence, we have

for each . Since is nonnegative, by Proposition 3.4 [25], we have

for each . This, together with equality (30), follows that

This completes the proof.

Theorem 9. Let be a set-valued measure and let be (KL) -integrable. Then

for each .

Proof. Since is (KL) -integrable, there exists such that

for each and . From the properties of support functions, we have

where is the total variation measure of . Note that satisfies

for each and with , where the supremum is taken over all finite measurable partitions of . Since is nonnegative, we have

for each and with . It follows that

for each . This completes the proof.

Corollary 10. Let be a set-valued measure and let be (KL) -integrable. Then

(1);(2)if , then ;(3)if , then for each .

Proof. (1) By Theorem 9, we have

which implies

Thus, .
(2) If , then . This implies

that is, . Since is a complete metric space, we have .
(3) If , then

for each and , which implies that for each . This completes the proof.

In the following, we give the Vitali type convergence theorem for the set-valued Kluvánek-Lewis integral.

Theorem 11. Let be a set-valued measure and let ,  , be (KL) -integrable such that is uniformly integrable with respect to and -a.e. Then

Proof. Since , are (KL) -integrable, there exist , ,  , such that

for each and . Then, by the properties of support functions, we have

In the same manner as in the proof of Theorem 9, we have

Then, by classical Vitali’s convergence theorem, we can conclude that

as . This completes the proof.

Similarly, we can obtain the Lebesgue type dominated convergence theorem for the set-valued Kluvánek-Lewis integral as follows.

Theorem 12. Let be a set-valued measure and let ,  , be (KL) -integrable such that -a.e. If there exists a nonnegative, -integrable function such that for all and , then

Proof. By Theorem 9 and classical Lebesgue dominated convergence theorem, we can obtain the result.

In the last part of this section, we discuss the relationship between set-valued Bartle integrals and set-valued Kluvánek-Lewis integrals. For convenience, we recall from [13] the definition of the set-valued Bartle integral as follows.

Definition 13. Let be a set-valued measure of bounded variation and let be an element of . One defines the integral of with respect to , which one calls the set-valued Bartle integral, as follows:

The vector-valued integrals of the right-hand side are defined in the sense of Bartle et al. [14].

In order to distinguish the set-valued Bartle integral from the set-valued Kluvánek-Lewis integral, in the sequel we denote the set-valued Bartle integral by .

Theorem 14. Suppose that has the RNP, is separable, is a set-valued measure of bounded variation, and is an element of . Then is (KL) -integrable and

Proof. Since has the RNP, is separable and is a set-valued measure of bounded variation; and satisfy all the conditions of Theorem 6.5.4 [9]. Then, by Corollary 6.5.2 of Theorem 6.5.4 [9], we have

for each and, by Corollary 6.5.1 of Theorem 6.5.4 [9], we have

It follows that is (KL) -integrable and

This completes the proof.

Remark 15. In the theory of vector-valued measures and integrals, Lewis ([23], Theorem 2.4) proves, using Vitali’s convergence theorem, that Kluvánek-Lewis integral is equivalent to the one given by Bartle et al. [14]. In that case, the set-valued Bartle integral coincides with the set of Kluvánek-Lewis integrals of with respect to measure selections of ; that is,

Thus, under the assumptions of Theorem 14, we can obtain that

4. Conclusion

In the current paper we introduce a new integral of scalar-valued functions with respect to a set-valued measure. We discuss its properties and establish convergence theorems. In all applications which involve measure and integral, when measurement or data are set-valued, the structure defined in this paper can be applied. There are several directions for further investigation connected with this topic: specific properties of the integral, application on random case, expectation and conditional expectation, and application in economy.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The project is supported by the National Natural Science Foundation of China (11371002, 41201327), Specialized Research Fund for the Doctoral Program of Higher Education (20131101110048), Natural Science Foundation of Hebei Province (A2013201119), and Science and Technology Bureau of Baoding City (14ZF058).

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