Lebesgue Decomposition Theorem and Weak Radon-Nikodým Theorem for Generalized Fuzzy Number Measures

Abstract

The Lebesgue type decomposition theorem and weak Radon-Nikodým theorem for fuzzy valued measures in separable Banach spaces are established.

1. Introduction

The topic of set valued and fuzzy valued measures has received much attention in the past few decades because of its usefulness in several applied fields like mathematical economics and optimal control. Significant contributions to set valued measures were made by Artstein [1], Cascales et al. [2, 3], Hiai [4], Papageorgiou [5], Stojaković [6], Zhang et al. [7], Zhou and Shi [8], and others. Fuzzy valued measure is a natural generalization of set valued measures. Hence the study of fuzzy valued measures is usually connected with set valued measures. Contributions in this field were made, among others, by Puri and Ralescu [9], Bán [10], Stojaković [11, 12], M. Stojaković and Z. Stojaković [13, 14], Xue et al. [15], and Park [16].

The main results of this paper fall into two main parts. It is well known that Lebesgue type decomposition theorem and Radon-Nikodým theorem are very important results in measure theory. As an extension of Lebesgue type decomposition theorem for vector measures, Zhang et al. [7] obtained Lebesgue type decomposition theorem for set valued measures. In the first part we generalize these results to generalized fuzzy number measures. On the other hand, Wu et al. [17] obtained Radon-Nikodým theorem for generalized fuzzy number measures using Bochner integral. But it is well known that Pettis integrability is a more general concept than that of Bochner integrability in the theory of integration in infinite dimensional spaces. In the second part, we obtain weak Radon-Nikodým theorem for generalized fuzzy number measures using Pettis type integral.

The paper is structured as follows. In Section 2, we state some basic concepts and preliminary results. In Section 3, the Lebesgue type decomposition theorem and weak Radon-Nikodým theorem for generalized fuzzy number measures will be established.

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 Note that is a complete metric space. The number is defined by for each

We will denote by the support function of a set defined by The support function satisfies the following properties: and for all and .

Definition 1 (see [15]). Let . One denotes for each . is called a generalized fuzzy number if, for each , . Let denote the set of all generalized fuzzy numbers on .

For , we define as follows: Obviously, we have for each , and therefore In the set we define by is a metric space.

Theorem 2 (see [18]). If , then one has the following:(1) for all .(2) for .(3)If is a nondecreasing sequence in converging to , then .Conversely, if satisfies (1)–(3) above, then there exists a such that for each

Theorem 3 (see [19]). Let , , and , ; then converges to for each if and only if .

Definition 4 (see [4]). Let be a measurable space. The mapping is said to be a set valued measure if it satisfies the following two conditions:(1);(2)if are in , with for , then  where

Particularly, is a set valued measure if and only if is a real valued measure for all .

As for single valued measures, we have the notion of total variation of . For we define , where the supremum is taken over all finite measurable partitions of . We call that is of bounded variation if . We call that is of -bounded variation if there exists a countably measurable partition such that each restriction of to is of bounded variation. We call that is continuous absolutely about if, for any , ; then , denoted as . We call that is singular about if there exists , , such that, for any , , denoted as .

Definition 5 (see [15]). Let be a measurable space. The mapping is called a generalized fuzzy number measure if it satisfies the following two conditions:(1), where is indicator function of .(2)If are in , with for , then  where

We call that is continuous absolutely about if, for any , ; then , denoted as .

Theorem 6 (see [15]). The mapping is a generalized fuzzy number measure if and only if there exists a family of set valued measures satisfying the following three conditions:(1)For arbitrary and , if , then .(2)For arbitrary and such that and , then .(3)For arbitrary , one has Note that for a generalized fuzzy number measure the set valued measure is determined by that is,

3. Main Results

In this section, we first give the Lebesgue type decomposition theorem for generalized fuzzy number measures. Our result is a generalization of Zhang et al.’s result [7]. And then, we obtain weak Radon-Nikodým theorem for generalized fuzzy number measures.

Theorem 7 (see [7] Lebesgue type decomposition theorem for set valued measures). Let be a set valued measure. Then there exists a unique pair of weakly compact and convex set valued measures and such that(1), ;(2);(3)for arbitrary , is the Lebesgue decomposition of

In what follows, we first give the definition of singular for generalized fuzzy number measure about crisp measure and give two lemmas before we establish the main result.

Definition 8. Let ) be a finite real valued measure space and let be a generalized fuzzy number measure. One calls that is singular about if there exists , , such that, for any , , denoted as .

Lemma 9. Let be a finite real valued measure space and let be a generalized fuzzy number measure. Then(1) if and only if for all ;(2) if and only if for all .

Proof. (1) Suppose that . For any , if then , which implies that for all ; that is, for all .
On the contrary, suppose that for all For any , if then . If , then there exists such that , which leads to the conclusion that there exists such that . This contradicts the fact that for all . Hence, . This implies that .
(2) Suppose that . If there exists , , then, for any , which implies that for all . It follows that for all .
Conversely, suppose that for all . According to Definition 8, if there exists , , then, for any and , we have , which implies that . Hence, .

Lemma 10. Let such that and let be a family of weakly compact and convex set valued measure such that for any . Then one has the following:(1) is a real valued measure for each .(2)If for each , then for each .(3)If for each , then for each .

Proof. (1) Since is a set valued measure for any , is a real valued measure for each and any . Further, implies for each and , which shows that is a monotone decreasing and bounded sequence for each and . This ensures that exists. In the following, we claim that is a real valued measure for each . Obviously, Let be a sequence of pairwise disjoint elements of . By countable additivity of set valued measure and Theorem 6.1.1 [7], we have for each . Sine for each and , we havewhich implies that This shows the countable additivity of for each .
(2) Suppose that for each . If, for , , then which implies that for each . It follows that for each . Thus for each
(3) Suppose that for each . If there exists , , then, for arbitrary , which implies that for all It follows that for each . Thus for each .

In the following, we give the Lebesgue type decomposition theorem for generalized fuzzy number measures.

Theorem 11. Let be a generalized fuzzy number measure. Then there exists a unique pair of generalized fuzzy number measures and such that

Proof. Firstly, we show the existence of decomposition. If is a generalized fuzzy number measure, then, by Theorem 6, defined by is a set valued measure for each . It follows from Theorem 7 that there exists a unique pair of weakly compact and convex set valued measures and such that for each . Further,is the Lebesgue decomposition of for any and .
To complete the proof, by Theorem 6, we show that and define two generalized fuzzy number measures and such that From (18), we havefor each , , and . Since is a generalized fuzzy number measure, by Theorem 6, for arbitrary with and , we have . It follows that for each and if . According to the uniqueness of Lebesgue decomposition for real valued measures, we have for each and , which imply for each Now let be a nondecreasing sequence in converging to . We use Theorem 3 to show that for any . By Theorem 7, we can conclude that there exists a unique pair of weakly compact and convex set valued measures and such that for any . It follows thatfor any and . Lemma 10 ensures that exist. Then, from (25), we haveAgain by Theorem 6, we havefor any . It follows from Theorem 3 thatfor each Then (27) and (29) imply relationfor each and . Lemma 10 shows that are two real valued measures such thatThen, by uniqueness of Lebesgue decomposition for real valued measure, (20) and (30) imply for any It follows from Theorem 3 that for each . Then and satisfy conditions (1) and (2) of Theorem 6, respectively. For any , let where . Then and define two generalized fuzzy number measures such that for any and . Since , for all , by Lemma 10, and .
In the following, we claim that . Obviously, for any , and for any ,Then, if , If , then , which implies that, for any , . Hence, for any , ; that is, , which implies that . It follows that .
Lastly, we show the uniqueness of decomposition. Suppose that there exist generalized fuzzy number valued measures , , such thatThen, for any , According to the uniqueness of Lebesgue decomposition of set valued measures, we havewhich implies that , This completes the proof.

In the following, we generalize weak Radon-Nikodým theorem for set valued measures [20] to generalized fuzzy number measures.

Let be a set valued function. is said to be scalarly measurable if, for each , the real valued function is measurable. is said to be weakly integrable bounded if the real valued function defined by is integrable for each . is called a measurable selector of if for each A measurable selector of is called a Pettis integrable selector of if is Pettis integrable. We denote by the set of all Pettis integrable selectors of . is said to be Aumann-Pettis integrable if . In this case, the Aumann-Pettis integral of is defined by We denote by the set of all Aumann-Pettis integrable, weakly compact, and convex set valued functions.

Theorem 12 (see [20]). Suppose that has the RNP and is separable. Let be a -continuous set valued measure of bounded variation. has a weak Radon-Nikodým derivative if and only if is of -bounded variation.

The mapping is called a measurable fuzzy mapping, if is a measurable set valued mapping for each . is called weakly integrable bounded if is weakly integrable bounded set valued mapping for each .

Definition 13 (see [19]). The mapping is said to be weakly integrable, if, for each , there exist such that, for each , In this case is called the weak integral of over .
Let denote the set of all weakly integrable bounded measurable fuzzy mappings.

Definition 14. Let be a generalized fuzzy number measure. If there exists a weakly integrable bounded measurable fuzzy mapping such that then one calls that is a weak Radon-Nikodým derivative for with respect to .
We call that is of -bounded variation if is of -bounded variation for each In the following we show the existence of weak Radon-Nikodým derivatives for generalized fuzzy number measures.

Theorem 15. Suppose that has the RNP and is separable. Let be a -continuous generalized fuzzy number measure of -bounded variation. Then has a weak Radon-Nikodým derivative ; namely, there exists a weakly integrable bounded measurable fuzzy mapping such that for each .

Proof. Since is a generalized fuzzy number measure, by Theorem 6, defined by is a set valued measure for each . Since is -continuous and is of -bounded variation, is -continuous and is of -bounded variation for each . According to Theorem 12, for each there exists which is a weak Radon-Nikodým derivative for with respect to ; namely, satisfiesfor each and .
In the following, we will show that the family defines a weakly integrable bounded measurable fuzzy mapping . Firstly, we prove that the family defines a generalized fuzzy number for each We will prove that all the conditions of Theorem 2 are satisfied. Obviously, for all and . For , we have for any which implies that Then, by properties of support function, we havefor each . It follows from Lemma 4.3 [19] that for each . Since this is true for all we deduce that Also, since and are weakly compact and convex, we have . Let be a nondecreasing sequence converging to . By Theorem 6, we have which implies that for each . It follows from Theorem 3 that By Lemma 4.3 [19], we haveSince is weak integrable bounded and for all , , and , by Lebesgue’s dominated convergence theorem and (54), we obtain that which implies thatAgain, by Theorem 3, we have . Now, applying Theorem 2, we get that the family generates the unique generalized fuzzy number such that for each Since is weakly integrable bounded set valued measurable function for each , is a weakly integrable bounded measurable fuzzy valued function.
To prove that is a weak Radon-Nikodým derivative for with respect to , we will show that for each By Definition 13, such that for each and . Together with (47), we have for each and , which implies that for each . This completes the proof.

Conflict of Interests

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

Acknowledgments

The project is supported by the National Natural Science Foundation of China (11371002, 41201327, and 61572011), Youth Scientific Research Foundation of Education Department of Hebei Province (QN2015005, QN2015026, and QN20131055), Specialized Research Fund for the Doctoral Program of Higher Education (20131101110048), Natural Science Foundation of Hebei Province (A2013201119), and Special Fund for Enhancing Comprehensive Strength of Midwest China.