Journal of Function Spaces
Volume 2015 (2015), Article ID 576134, 8 pages
http://dx.doi.org/10.1155/2015/576134

Research Article

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

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 17 September 2015; Accepted 27 October 2015

Academic Editor: Luisa  Di Piazza

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.

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 have

which 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 have

for 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 that

for any and . Lemma 10 ensures that

exist. Then, from (25), we have

Again by Theorem 6, we have

for any . It follows from Theorem 3 that

for each Then (27) and (29) imply relation

for each and . Lemma 10 shows that

are two real valued measures such that

Then, 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 that

Then, for any ,

According to the uniqueness of Lebesgue decomposition of set valued measures, we have

which 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, satisfies

for 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 have

for 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 have

Since is weak integrable bounded and

for all , , and , by Lebesgue’s dominated convergence theorem and (54), we obtain that

which implies that

Again, 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.

References

Z. Artstein, “Set-valued measures,” Transactions of the American Mathematical Society, vol. 165, pp. 103–125, 1972.

B. Cascales, V. Kadets, and J. Rodríguez, “Measurable selectors and set-valued Pettis integral in non-separable Banach spaces,” Journal of Functional Analysis, vol. 256, no. 3, pp. 673–699, 2009.

B. Cascales, V. Kadets, and J. Rodríguez, “Radon-Nikodým theorems for multimeasures in non-separable spaces,” Journal of Mathematical Physics, Analysis, Geometry, vol. 9, no. 1, pp. 7–24, 2013.

F. Hiai, “Radon-Nikodym theorems for set-valued measures,” Journal of Multivariate Analysis, vol. 8, no. 1, pp. 96–118, 1978.

N. S. Papageorgiou, “Contributions to the theory of set valued functions and set valued measures,” Transactions of the American Mathematical Society, vol. 304, no. 1, pp. 245–245, 1987.

M. Stojaković, “Set valued probability and its connection with set valued measure,” Statistics & Probability Letters, vol. 82, no. 6, pp. 1043–1048, 2012.

W. Zhang, S. Li, Z. Wang, and Y. Gao, Set-Valued Stochastic Process, Academic Press, Beijing, China, 2007.

C.-L. Zhou and F.-G. Shi, “New set-valued integral in a Banach space,” Journal of Function Spaces, vol. 2015, Article ID 260238, 8 pages, 2015.

M. L. Puri and D. A. Ralescu, “Convergence theorem for fuzzy martingales,” Journal of Mathematical Analysis and Applications, vol. 160, no. 1, pp. 107–122, 1991.

J. Bán, “Radon-Nikodým theorem and conditional expectation of fuzzy-valued measures and variables,” Fuzzy Sets and Systems, vol. 34, no. 3, pp. 383–392, 1990.

M. Stojaković, “Decomposition and representation of fuzzy valued measure,” Fuzzy Sets and Systems, vol. 112, no. 2, pp. 251–256, 2000.

M. Stojaković, “Imprecise set and fuzzy valued probability,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4524–4531, 2011.

M. Stojaković and Z. Stojaković, “Integral with respect to fuzzy measure in finite dimensional Banach spaces,” Novi Sad Journal of Mathematics, vol. 37, no. 1, pp. 163–170, 2007.

M. Stojaković and Z. Stojaković, “Fuzzy valued measure based integral,” Novi Sad Journal of Mathematics, vol. 37, no. 2, pp. 207–220, 2007.

X. Xue, M. Ha, and C. Wu, “On the extension of the fuzzy number measures in Banach spaces: part I. Representation of the fuzzy number measures,” Fuzzy Sets and Systems, vol. 78, no. 3, pp. 347–354, 1996.

C.-K. Park, “Generalized fuzzy number valued Bartle integrals,” Communications of the Korean Mathematical Society, vol. 25, no. 1, pp. 37–49, 2010.

J. Wu, X. Xue, and C. Wu, “Radon-Nikodym theorem and Vitali-Hahn-Saks theorem on fuzzy number measures in Banach spaces,” Fuzzy Sets and Systems, vol. 117, no. 3, pp. 339–346, 2001.

J. Wu and C. Wu, “The w-derivatives of fuzzy mappings in Banach spaces,” Fuzzy Sets and Systems, vol. 119, no. 3, pp. 375–381, 2001.

X. P. Xue, M. H. Ha, and M. Ma, “Random fuzzy number integrals in Banach spaces,” Fuzzy Sets and Systems, vol. 66, no. 1, pp. 97–111, 1994.

J. R. Wu and C. X. Wu, “Existence of weak Radon-Nikodým derivatives for set-valued measures and conditional expectations for weak set-valued random variables,” Acta Mathematicae Applicatae Sinica, vol. 24, no. 2, pp. 255–261, 2001.