Journal of Applied Mathematics

Journal of Applied Mathematics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 492149 | https://doi.org/10.1155/2011/492149

Lee-Chae Jang, "On Properties of the Choquet Integral of Interval-Valued Functions", Journal of Applied Mathematics, vol. 2011, Article ID 492149, 10 pages, 2011. https://doi.org/10.1155/2011/492149

On Properties of the Choquet Integral of Interval-Valued Functions

Academic Editor: F. Marcellán
Received16 Jul 2011
Accepted06 Sep 2011
Published29 Oct 2011

Abstract

Based on the concept of an interval-valued function which is motivated by the goal to represent an uncertain function, we define the Choquet integral with respect to a fuzzy measure of interval-valued functions. We also discuss convergence in the mean and convergence in a fuzzy measure of sequences of measurable interval-valued functions. In particular, we investigate the convergence theorem for the Choquet integral of measurable interval-valued functions.

1. Introduction

Wang [1], Pedrycz et al. [2], Ha and Wu [3], and T. Murofushi et al. [4, 5] defined the concepts of various convergence of sequences of measurable functions and discussed its theoretical underpinnings along with related interpretation issues. Many researchers [2, 411] also have been studying the Choquet integral which is regarded as one of aggregation operator being used in the decision making and information theory.

The main idea of this study is the concept of interval-valued functions which is associated with the representation of uncertain functions. In the past decade, it has been suggested to use intervals in order to represent uncertainty, for examples, closed set-valued functions [4, 713], interval-valued probability [14], fuzzy set-valued measures [13], and economic uncertainty [14].

In Section 2, we list definitions and basic properties of a fuzzy measure, the Choquet integral, and various convergences of sequences of measurable functions. In Section 3, we provide the new definitions of the Choquet integral with respect to a fuzzy measure of measurable interval-valued functions as well as various convergences of sequences of measurable interval-valued functions and investigate their properties. We also discuss convergence in the mean and convergence in a fuzzy measure of sequences of measurable interval-valued functions. In particular, we prove the convergence theorem for the Choquet integral of measurable interval-valued functions. In Section 4, we give a brief summary results and some conclusions.

2. Preliminaries and Definitions

Let be a measurable space, where denote a nonempty set, and stands for a -algebra of subsets of . Denote by the set of all nonnegative measurable functions on , , and .

Definition 2.1 (see [15]). A set function is called a fuzzy measure if(FM1) (vanishes on );(FM2) and (monotonicity);(FM3) (continuity from below);(FM4) and (continuity from above).

Remark that a fuzzy measure is known to be the generalization of a classical measure satisfying (FM3) and (FM4) where additivity is replaced by the weaker condition of monotonicity.

Definition 2.2 (see [5]). (1) A fuzzy measure is said to be autocontinuous from above (resp., below) if , , and (resp., ).
(2) If is autocontinuous both from above and from below, it is said to be autocontinuous.

Definition 2.3 (see [1, 2, 4, 5]). (1) Let and . The Choquet integral of with respect to a fuzzy measure is defined by where the integral on the right-hand side is the Lebesgue integral and .
(2) A measurable function is said to be integrable if the Choquet integral of on exists and its value is finite.

Instead of , we write . Note that if we take and is the power set of and is a measurable function on , then where is a permutation on such that and and . From (2.2), clearly we have

Definition 2.4 (see [1, 2]). Let and . A sequence converges in the mean to if

Definition 2.5 (see [1, 2]). Let . A sequence is called equally integrable on if for any given , there exists such that for all .

It is easy to see that if there exists a integrable function such that for all , then is equally integrable.

Definition 2.6 (see [2]). Let , , and . We say that converges in to on if for any given ,

Definition 2.7 (see [2, 4, 5]). Let . and are comonotonic if for every pair ,

Theorem 2.8 (see [15]). Let , , and be a fuzzy measure.
(1) If , then .
(2) If and are nonnegative real numbers, then
(3) If and are comonotonic, then
(4) If we define for all , then
(5) If we define for all , then

Theorem 2.9 (see [1, 2]). Let and be equally integrable. If converges in the mean to and is autocontinuous, then

Note that if satisfies (2.12), then it is said to be Choquet weak converge to .

3. Interval-Valued Functions and the Choquet Integral

Let be the set of all closed subsets in and the set of all bounded closed intervals (intervals, for short) in , that is, For any , we define . Obviously, (see [9, 10, 14, 15]).

Definition 3.1. If , , and , then we define arithmetic, maximum, minimum, order, and inclusion operations as follows:(1),(2),(3),(4),(5),(6) if and only if and ,(7) if and only if and , and(8) if and only if and .

Let be the set of all measurable closed set-valued functions on and the set of all measurable interval-valued functions. Recall that a closed set-valued function is said to be measurable if for any open set , Then, we introduce the Choquet integral of measurable interval-valued functions (see [711]).

Definition 3.2 (see [11]). (1) Let and be a fuzzy measure. The Choquet integral of with respect to on is defined by where is the family of measurable selections of , that is,
(2) is said to be integrable if .
(3) is said to be integrably function such that

We note that -a.e. means almost everywhere in a fuzzy measure . Then, we obtain the following theorem which is a useful tool to investigate various convergences of interval-valued functions.

Theorem 3.3 (see [11, Theorem  3.16(iii)]). Let and be a fuzzy measure. If is a integrably bounded interval-valued function, then

Now, we define convergence in the mean, equally integrable, and convergence in a fuzzy measure and discuss their properties.

Definition 3.4. Let and . A sequence converges in the mean to if where is the Hausdorrf metric on .

Recall that for each pair , Then, it is easy to see that for each pair , , By (3.7) and (3.9), we obtain the following theorem.

Theorem 3.5. Let and . If a sequence converges in the mean to , then a sequence (resp., ) converges in the mean to (resp., ).

Proof. By (3.7) and Theorem 2.8 (4), we have From (3.10), we can derive the followings: Thus, the proof is complete.

Definition 3.6. Let . A sequence is called equally integrable on if for any given , there exists such that for all .

Theorem 3.7. Let and be a fuzzy measure. If a sequence is equally integrable on , then sequences and are equally integrable on .

Proof. By (3.9) and (3.12), we have By the assumption and (3.13), given , there exists such that Thus we have Thus, the proof is complete.

Definition 3.8. Let , and . We say that converges in to on if for any given ,

Theorem 3.9. Let , and . If converges in to on , then a sequence (resp., ) converges in to (resp., ) on .

Proof. By (3.9), we have Since and , by (3.16), we have Thus, the proof is complete.

Definition 3.10. Let . and are comonotonic if and only if for every pair ,

Theorem 3.11. Let , . If and are comonotonic, then and are comonotonic, and and are comonotonic.

Proof. If and and , then we have . Since and are comonotonic, . Then, we have and . Thus, the proof is complete.

Then, we discuss some properties of the Choquet integral with respect to a fuzzy measure of measurable interval-valued functions.

Theorem 3.12. Let , , and let be a fuzzy measure.
(1) If , then .
(2) If and are nonnegative real numbers, then
(3) If and are comonotonic, then
(4) If we define for all , then
(5) If we define for all , then

Proof. (1) If , then and . By Theorem 2.8(1), and . By Definition 3.1(6) and (3.6), we have
(2) By the same method of (1) and Theorem 2.8(2), we have
(3) The proof is similar to (2).
(4) Since , by Theorem 2.8(4),
(5) The proof is similar to (4).

Finally, we obtain the main result which is the following convergence theorem for the Choquet integral with respect to an autocontinuous fuzzy measure of a measurable interval-valued function.

Theorem 3.13. If and is equally integrable and If converges in the to and is autocontinuous, then

Proof. Since is equally integrable on , by Theorem 3.7, and are equally integrable on . By Theorem 2.9, Thus, for any given , there exists such that for all . Then, we have for all . Thus, the proof is complete.

4. Conclusions

In this paper, we consider the new concept of the Choquet integral of a measurable interval-valued function which generalizes the Choquet integral of a measurable function mentioned in the papers [26, 11]. From Theorems 3.12 and 3.13, we established fundamental properties of the Choquet integral of interval-valued functions and the convergence theorem for the Choquet integral with respect to an autocontinuous fuzzy measure of measurable interval-valued functions.

In the future, by using these results of this paper, we can develop various problems, for example, the Choquet weak convergence of uncertain random sets and the weak convergence theorems for the Aumann integral of measurable interval-valued functions, and so forth.

Acknowledgment

This paper was supported by Konkuk University in 2011.

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Copyright © 2011 Lee-Chae Jang. 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.


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