On Properties of the Choquet Integral of Interval-Valued Functions

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 (𝐶) 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, 4–11] 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, 7–13], 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 (𝐶) 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 (𝑋,𝔅) be a measurable space, where 𝑋 denote a nonempty set, and 𝔅 stands for a 𝜎-algebra of subsets of 𝑋. Denote 𝔉 by the set of all nonnegative measurable functions on (𝑋,𝔅), 𝑅⁺=[0,∞), and 𝑅⁺=[0,∞].

Definition 2.1 (see [1–5]). (1) A set function 𝜇∶𝔅→𝑅⁺ is called a fuzzy measure if(FM1)𝜇(∅)=0 (vanishes on ∅);(FM2)𝐴,𝐵∈𝔅 and 𝐴⊂𝐵⇒𝜇(𝐴)≤𝜇(𝐵) (monotonicity);(FM3)𝐴₁⊂𝐴₂⊂⋯⊂⋯,𝐴_𝑛∈𝔅(𝑛=1,2,…)⇒lim_𝑛→∞𝜇(𝐴_𝑛)=𝜇(∪^∞_𝑛=1𝐴_𝑛) (continuity from below);(FM4)𝐴₁⊃𝐴₂⊃⋯⊃𝐴_𝑛⊃⋯,𝐴_𝑛∈𝔅(𝑛=1,2,…) and 𝜇(𝐴₁)<∞⇒lim_𝑛→∞𝜇(𝐴_𝑛)=𝜇(∩^∞_𝑛=1𝐴_𝑛) (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 𝜇 is said to be autocontinuous from above (resp., below) if 𝐴∈𝔉, {𝐵_𝑛}⊂𝔉, and lim_𝑛→∞𝜇(𝐵_𝑛)=0⇒lim_𝑛→∞𝜇(𝐴∪𝐵_𝑛)=𝜇(𝐴) (resp., lim_𝑛→∞𝜇(𝐴⧵𝐵_𝑛)=𝜇(𝐴)).
(2) If 𝜇 is autocontinuous both from above and from below, it is said to be autocontinuous.

Definition 2.3 (see [1, 2, 4, 5]). (1) Let 𝐴∈𝔅 and 𝑓∈𝔉. The Choquet integral of 𝑓 with respect to a fuzzy measure 𝜇 is defined by (𝐶)_𝐴𝑓𝑑𝜇=^∞₀𝜇𝐺_𝑓(𝛼)∩𝐴𝑑𝛼, (2.1) where the integral on the right-hand side is the Lebesgue integral and 𝐺_𝑓(𝛼)={𝑥∣𝑓(𝑥)≥𝛼}.
(2) A measurable function 𝑓 is said to be (𝐶) integrable if the Choquet integral of 𝑓 on 𝑋 exists and its value is finite.

Instead of (𝐶)∫_𝑋𝑓𝑑𝜇, we write (𝐶)∫𝑓𝑑𝜇. Note that if we take 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} and 𝔅=℘(𝑋) is the power set of 𝑋 and 𝑓 is a measurable function on 𝑋, then (𝐶)_𝐴𝑓𝑑𝜇=^𝑛_𝑖=1𝑓𝑥_(𝑖)𝜇𝐴_(𝑖)−𝜇𝐴_(𝑖+1), (2.2) where (⋅) is a permutation on {1,2,…,𝑛} such that 0=𝑓(𝑥₍₀₎)≤𝑓(𝑥₍₁₎)≤⋯≤𝑓(𝑥_(𝑛)) and 𝐴_(𝑖)={𝑥_(𝑖),𝑥_(𝑖+1),…,𝑥_(𝑛)} and 𝐴_(𝑛+1)=∅. From (2.2), clearly we have (𝐶)_𝐴𝑓𝑑𝜇=^𝑛_𝑖=1𝑓𝑥_(𝑖)−𝑓𝑥_(𝑖−1)𝜇𝐴_(𝑖). (2.3)

Definition 2.4 (see [1, 2]). Let {𝑓_𝑛}⊂𝔉 and 𝑓∈𝔉. A sequence {𝑓_𝑛} converges in the (𝐶) mean to 𝑓 if lim_𝑛→∞(𝐶)_𝐴𝑓_𝑛−𝑓𝑑𝜇=0. (2.4)

Definition 2.5 (see [1, 2]). Let 𝐴∈𝔅. A sequence {𝑓_𝑛} is called equally (𝐶) integrable on 𝐴 if for any given 𝜀>0, there exists 𝑁(𝜀)>0 such that (𝐶)_𝐴𝑓_𝑛𝑑𝜇≤^𝑁₀𝜇𝐺_𝑓𝑛(𝛼)∩𝐴𝑑𝛼+𝜀 (2.5) for all 𝑛=1,2,….

It is easy to see that if there exists a (𝐶) integrable function 𝑔 such that |𝑓_𝑛|≤𝑔 for all 𝑛=1,2,…, then {𝑓_𝑛} is equally (𝐶) integrable.

Definition 2.6 (see [2]). Let 𝐴∈𝔅, {𝑓_𝑛}⊂𝔉, and 𝑓∈𝔉. We say that {𝑓_𝑛} converges in 𝜇 to 𝑓 on 𝐴 if for any given 𝜀>0, lim_𝑛→∞𝜇𝑥∣𝑓_𝑛(𝑥)−𝑓(𝑥)≥𝜀∩𝐴=0. (2.6)

Definition 2.7 (see [2, 4, 5]). Let 𝑓,𝑔∈𝔉. 𝑓 and 𝑔 are comonotonic if for every pair 𝑥,𝑦∈𝑋, 𝑓(𝑥)<𝑓𝑦⟹𝑔(𝑥)≤𝑔𝑦. (2.7)

Theorem 2.8 (see [1–5]). Let 𝐴∈𝔅, 𝑓,𝑔∈𝔉, and 𝜇 be a fuzzy measure.
(1) If 𝑓≤𝑔, then (𝐶)∫_𝐴𝑓𝑑𝜇≤(𝐶)∫_𝐴𝑔𝑑𝜇.
(2) If 𝑎 and 𝑏 are nonnegative real numbers, then (𝐶)_𝐴𝑎𝑓+𝑏𝑑𝜇=𝑎(𝐶)_𝐴𝑓𝑑𝜇+𝑏𝜇(𝐴). (2.8)
(3) If 𝑓 and 𝑔 are comonotonic, then (𝐶)_𝐴𝑓+𝑔𝑑𝜇=(𝐶)_𝐴𝑓𝑑𝜇+(𝐶)_𝐴𝑔𝑑𝜇. (2.9)
(4) If we define (𝑓∨𝑔)(𝑥)=𝑓(𝑥)∨𝑔(𝑥) for all 𝑥∈𝑋, then (𝐶)_𝐴𝑓∨𝑔𝑑𝜇≥(𝐶)_𝐴𝑓𝑑𝜇∨(𝐶)_𝐴𝑔𝑑𝜇. (2.10)
(5) If we define (𝑓∧𝑔)(𝑥)=𝑓(𝑥)∧𝑔(𝑥) for all 𝑥∈𝑋, then (𝐶)_𝐴𝑓∧𝑔𝑑𝜇≤(𝐶)_𝐴𝑓𝑑𝜇∧(𝐶)_𝐴𝑔𝑑𝜇. (2.11)

Theorem 2.9 (see [1, 2]). Let 𝐴∈𝔅 and {𝑓_𝑛} be equally (𝐶) integrable. If {𝑓_𝑛} converges in the (𝐶) mean to 𝑓 and 𝜇 is autocontinuous, then lim_𝑛→∞(𝐶)_𝐴𝑓_𝑛𝑑𝜇=(𝐶)_𝐴𝑓𝑑𝜇. (2.12)

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

3. Interval-Valued Functions and the Choquet Integral

Let 𝐶(𝑅⁺) be the set of all closed subsets in ℝ⁺ and 𝐼(ℝ) the set of all bounded closed intervals (intervals, for short) in ℝ=(−∞,∞), that is, 𝐼(𝑅)=𝑎^𝑙,𝑎^𝑟∣𝑎^𝑙,𝑎^𝑟∈ℝ,𝑎^𝑙≤𝑎^𝑟. (3.1) For any 𝑎∈ℝ, we define 𝑎=[𝑎,𝑎]. Obviously, 𝑎∈𝐼(ℝ) (see [9, 10, 14, 15]).

Definition 3.1. If 𝑎=[𝑎^𝑙,𝑎^𝑟], 𝑏=[𝑏^𝑙,𝑏^𝑟]∈𝐼(ℝ⁺), and 𝑘∈ℝ⁺, then we define arithmetic, maximum, minimum, order, and inclusion operations as follows:(1)𝑎+𝑏=[𝑎^𝑙+𝑏^𝑙,𝑎^𝑟+𝑏^𝑟],(2)𝑘𝑎=[𝑘𝑎^𝑙,𝑘𝑎^𝑟],(3)𝑎𝑏=[𝑎^𝑙𝑏^𝑙,𝑎^𝑟𝑏^𝑟],(4)𝑎∨𝑏=[𝑎^𝑙∨𝑏^𝑙,𝑎^𝑟∨𝑏^𝑟],(5)𝑎∧𝑏=[𝑎^𝑙∧𝑏^𝑙,𝑎^𝑟∧𝑏^𝑟],(6)𝑎≤𝑏 if and only if 𝑎^𝑙≤𝑏^𝑙 and 𝑎^𝑟≤𝑏^𝑟,(7)𝑎<𝑏 if and only if 𝑎≤𝑏 and 𝑎≠𝑏, and(8)𝑎⊂𝑏 if and only if 𝑏^𝑙≤𝑎^𝑙 and 𝑎^𝑟≤𝑏^𝑟.

Let 𝔉_𝑐 be the set of all measurable closed set-valued functions on (𝑋,𝔅) and 𝔉_𝑖 the set of all measurable interval-valued functions. Recall that a closed set-valued function 𝐹∶𝑋→𝐶(ℝ⁺) is said to be measurable if for any open set 𝑂⊂ℝ⁺, 𝐹⁻¹(𝑂)=𝑥∈ℝ⁺∣𝐹(𝑥)∩𝑂≠∅∈𝔅. (3.2) Then, we introduce the Choquet integral of measurable interval-valued functions (see [7–11]).

Definition 3.2 (see [11]). (1) Let 𝐴∈𝔅,𝐹∈𝔉_𝑐 and 𝜇 be a fuzzy measure. The Choquet integral of 𝐹 with respect to 𝜇 on 𝐴 is defined by (𝐶)_𝐴𝐹𝑑𝜇=(𝐶)_𝐴𝑓𝑑𝜇∣𝑓∈𝑆_𝑐(𝐹), (3.3) where 𝑆_𝑐(𝐹) is the family of measurable selections of 𝐹, that is, 𝑆_𝑐(𝐹)=𝑓∶𝑋→ℝ⁺∣(𝐶)_𝐴𝑓𝑑𝜇<∞,𝑓(𝑥)∈𝐹(𝑥)𝜇−a.e.. (3.4)
(2) 𝐹 is said to be (𝐶) integrable if (𝐶)∫𝐹𝑑𝜇≠∅.
(3) 𝐹 is said to be (𝐶) integrably function 𝑓 such that 𝐹(𝑥)=sup_{(𝑥)𝑟∈𝐹}𝑟≤𝑔(𝑥)∀𝑥∈𝑋. (3.5)

We note that 𝜇-a.e. means almost everywhere in a fuzzy measure 𝜇. 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 𝐴∈𝔅 and 𝜇 be a fuzzy measure. If 𝐹=[𝑓^𝑙,𝑓^𝑟]∶𝑋→𝐼(𝑅⁺) is a (𝐶) integrably bounded interval-valued function, then (𝐶)𝐹𝑑𝜇=(𝐶)𝑓^𝑙𝑑𝜇,(𝐶)𝑓^𝑟𝑑𝜇. (3.6)

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

Definition 3.4. Let {𝐹_𝑛}⊂𝔉_𝑖 and 𝐹∈𝔉_𝑖. A sequence {𝐹_𝑛} converges in the (𝐶) mean to 𝐹 if lim_𝑛→∞(𝐶)𝑑_𝐻𝐹_𝑛(𝑥),𝐹(𝑥)𝑑𝜇=0, (3.7) where 𝑑_𝐻 is the Hausdorrf metric on 𝐶(ℝ⁺).

Recall that for each pair 𝐴,𝐵∈𝐶(ℝ⁺), 𝑑_𝐻(𝐴,𝐵)=maxsup_𝑦∈𝐵inf_𝑥∈𝐴𝑥−𝑦,sup_𝑥∈𝐴inf_𝑦∈𝐵𝑥−𝑦. (3.8) Then, it is easy to see that for each pair 𝑎=[𝑎^𝑙,𝑎^𝑟], 𝑏=[𝑏^𝑙,𝑏^𝑟]∈𝐼(ℝ⁺), 𝑑_𝐻𝑎,𝑏=max𝑎^𝑙−𝑏^𝑙,𝑎^𝑟−𝑏^𝑟. (3.9) By (3.7) and (3.9), we obtain the following theorem.

Theorem 3.5. Let {𝐹_𝑛=[𝑓^𝑙_𝑛,𝑓^𝑟_𝑛]}⊂𝔉_𝑖 and 𝐹=[𝑓^𝑙,𝑓^𝑟]∈𝔉. If a sequence {𝐹_𝑛} converges in the (𝐶) mean to 𝐹, then a sequence {𝑓^𝑙_𝑛} (resp., {𝑓^𝑟_𝑛}) converges in the (𝐶) mean to 𝑓^𝑙 (resp., 𝑓^𝑟).

Proof. By (3.7) and Theorem 2.8 (4), we have (𝐶)𝑑_𝐻𝐹_𝑛(𝑥)(𝑥)(𝑥)=(𝐶)𝑓,𝐹𝑑𝜇max^𝑙_𝑛(𝑥)−𝑓^𝑙(𝑥),𝑓^𝑟_𝑛(𝑥)−𝑓^𝑟(𝑥≥(𝐶)𝑓)𝑑𝜇_𝑙𝑛(𝑥)−𝑓^𝑙(𝑥)𝑑𝜇(𝑥)∨(𝐶)𝑓^𝑟_𝑛(𝑥)−𝑓^𝑟(𝑥)𝑑𝜇(𝑥). (3.10) From (3.10), we can derive the followings: lim_𝑛→∞(𝐶)𝑓^𝑙_𝑛−𝑓^𝑙𝑑𝜇=0,lim_𝑛→∞(𝐶)𝑓^𝑟_𝑛−𝑓^𝑟𝑑𝜇=0. (3.11) Thus, the proof is complete.

Definition 3.6. Let 𝐴∈𝔅. A sequence {𝐹_𝑛} is called equally (𝐶) integrable on 𝐴 if for any given 𝜀>0, there exists 𝑁(𝜀)>0 such that 𝑑_𝐻(𝐶)_𝐴𝐹_𝑛𝑑𝜇,^𝑁₀𝜇𝐺_𝑓^𝑙_𝑛(𝛼)∩𝐴𝑑𝛼,^𝑁₀𝜇𝐺_𝑓^𝑟_𝑛(𝛼)∩𝐴𝑑𝛼≤𝜀, (3.12) for all 𝑛=1,2,….

Theorem 3.7. Let 𝐴∈𝔅 and 𝜇 be a fuzzy measure. If a sequence {𝐹_𝑛} is equally (𝐶) integrable on 𝐴, then sequences {𝑓^𝑙_𝑛} and {𝑓^𝑟_𝑛} are equally (𝐶) integrable on 𝐴.

Proof. By (3.9) and (3.12), we have 𝑑_𝐻(𝐶)_𝐴𝐹_𝑛𝑑𝜇,^𝑁₀𝜇𝐺_𝑓^𝑙_𝑛(𝛼)∩𝐴𝑑𝛼,^𝑁₀𝜇𝐺_𝑓^𝑟_𝑛(𝛼)𝐶)∩𝐴𝑑𝛼=max(_𝐴𝑓^𝑙𝑑𝜇−^𝑁₀𝜇𝐺_𝑓^𝑙_𝑛(𝛼)∩𝐴𝑑𝛼,(𝐶)_𝐴𝑓^𝑟𝑑𝜇−^𝑁₀𝜇𝐺_𝑓^𝑟_𝑛(𝛼)∩𝐴𝑑𝛼. (3.13) By the assumption and (3.13), given 𝜀>0, there exists 𝑁(𝜀)>0 such that max(𝐶)_𝐴𝑓^𝑙𝑑𝜇−^𝑁₀𝜇𝐺_𝑓^𝑙_𝑛(𝛼)∩𝐴𝑑𝛼,(𝐶)_𝐴𝑓^𝑟𝑑𝜇−^𝑁₀𝜇𝐺_𝑓^𝑟_𝑛(𝛼)∩𝐴𝑑𝛼≤𝜀. (3.14) Thus we have (𝐶)_𝐴𝑓^𝑙𝑑𝜇−^𝑁₀𝜇𝐺_𝑓^𝑙_𝑛(𝛼)(𝐶)∩𝐴𝑑𝛼≤𝜀,_𝐴𝑓^𝑟𝑑𝜇−^𝑁₀𝜇𝐺_𝑓^𝑟_𝑛(𝛼)∩𝐴𝑑𝛼≤𝜀. (3.15) Thus, the proof is complete.

Definition 3.8. Let 𝐴∈𝔅,{𝐹_𝑛}⊂𝔉_𝑖, and 𝐹∈𝔉_𝑖. We say that {𝐹_𝑛} converges in 𝜇 to 𝐹 on 𝐴 if for any given 𝜀>0, lim_𝑛→∞𝜇𝑥∣𝑑_𝐻𝐹_𝑛(𝑥),𝐹(𝑥)≥𝜀∩𝐴=0. (3.16)

Theorem 3.9. Let 𝐴∈𝔅,{𝐹_𝑛=[𝑓^𝑙_𝑛,𝑓^𝑟_𝑛]}⊂𝔉_𝑖, and 𝐹=[𝑓^𝑙,𝑓^𝑟]∈𝔉_𝑖. If {𝐹_𝑛} converges in 𝜇 to 𝐹 on 𝐴, then a sequence {𝑓^𝑙_𝑛} (resp., {𝑓^𝑟_𝑛}) converges in 𝜇 to 𝑓^𝑙 (resp., 𝑓^𝑟) on 𝐴.

Proof. By (3.9), we have 𝑑_𝐻𝐹_𝑛(𝑥),𝐹(𝑥)=max𝑓^𝑙_𝑛(𝑥)−𝑓^𝑙(𝑥),𝑓^𝑟_𝑛(𝑥)−𝑓^𝑟(𝑥),∀𝑥∈𝑋. (3.17) Since {𝑥∣|𝑓^𝑙_𝑛(𝑥)−𝑓^𝑙(𝑥)|≥𝜀}⊂{𝑥∣𝑑_𝐻(𝐹_𝑛(𝑥),𝐹(𝑥))≥𝜀} and {𝑥|𝑓^𝑟_𝑛(𝑥)−𝑓^𝑟(𝑥)|≥𝜀}⊂{𝑥∣𝑑_𝐻(𝐹_𝑛(𝑥),𝐹(𝑥))≥𝜀}, by (3.16), we have lim_𝑛→∞𝜇𝑥∣𝑓^𝑙_𝑛(𝑥)−𝑓^𝑙(𝑥)≥𝜀∩𝐴=0,lim_𝑛→∞𝜇𝑥∣𝑓^𝑟_𝑛(𝑥)−𝑓^𝑟(𝑥)≥𝜀∩𝐴=0. (3.18) Thus, the proof is complete.

Definition 3.10. Let 𝐹,𝐺∈𝔉_𝑖. 𝐹 and 𝐺 are comonotonic if and only if for every pair 𝑥,𝑦∈𝑋, 𝐹(𝑥)<𝐹𝑦⟹𝐹(𝑥)≤𝐺𝑦. (3.19)

Theorem 3.11. Let 𝐹=[𝑓^𝑙,𝑓^𝑟], 𝐺=[𝑔^𝑙,𝑔^𝑟]∈𝔉_𝑖. If 𝐹 and 𝐺 are comonotonic, then 𝑓^𝑙 and 𝑔^𝑙 are comonotonic, and 𝑓^𝑟 and 𝑔^𝑟 are comonotonic.

Proof. If 𝑥,𝑦∈𝑋 and 𝑓^𝑙(𝑥)<𝑓^𝑙(𝑦) and 𝑓^𝑟(𝑥)<𝑓^𝑟(𝑦), then we have 𝐹(𝑥)<𝐺(𝑦). Since 𝐹 and 𝐺 are comonotonic, 𝐺(𝑥)≤𝐺(𝑦). Then, we have 𝑔^𝑙(𝑥)≤𝑔^𝑙(𝑦) and 𝑔^𝑟(𝑥)≤𝑔^𝑟(𝑦). 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 𝐴∈𝔅, 𝐹,𝐺∈𝔉_𝑖, and let 𝜇 be a fuzzy measure.
(1) If 𝐹≤𝐺, then (𝐶)∫_𝐴𝐹𝑑𝜇≤(𝐶)∫_𝐴𝐺𝑑𝜇.
(2) If 𝑎 and 𝑏 are nonnegative real numbers, then (𝐶)_𝐴(𝑎𝐹+𝑏)𝑑𝜇=𝑎(𝐶)_𝐴𝐹𝑑𝜇+𝑏𝜇(𝐴). (3.20)
(3) If 𝐹 and 𝐺 are comonotonic, then (𝐶)_𝐴(𝐹+𝐺)𝑑𝜇=(𝐶)_𝐴𝐹𝑑𝜇+(𝐶)_𝐴𝐺𝑑𝜇. (3.21)
(4) If we define (𝐹∨𝐺)(𝑥)=𝐹(𝑥)∨𝐺(𝑥) for all 𝑥∈𝑋, then (𝐶)_𝐴𝐹∨𝐺𝑑𝜇≥(𝐶)_𝐴𝐹𝑑𝜇∨(𝐶)_𝐴𝐺𝑑𝜇. (3.22)
(5) If we define (𝐹∧𝐺)(𝑥)=𝐹(𝑥)∧𝐺(𝑥) for all 𝑥∈𝑋, then (𝐶)_𝐴𝐹∧𝐺𝑑𝜇≤(𝐶)_𝐴𝐹𝑑𝜇∧(𝐶)_𝐴𝐺𝑑𝜇. (3.23)

Proof. (1) If 𝐹=[𝑓^𝑙,𝑓^𝑟]≤𝐺=[𝑔^𝑙,𝑔^𝑟], then 𝑓^𝑙≤𝑔^𝑙 and 𝑓^𝑟≤𝑔^𝑟. By Theorem 2.8(1), (𝐶)∫𝑓^𝑙𝑑𝜇≤(𝐶)∫𝑔^𝑙𝑑𝜇 and (𝐶)∫𝑓^𝑟𝑑𝜇≤(𝐶)∫𝑔^𝑟𝑑𝜇. By Definition 3.1(6) and (3.6), we have (𝐶)𝐹𝑑𝜇=(𝐶)𝑓^𝑙𝑑𝜇,(𝐶)𝑓^𝑟𝑑𝜇≤(𝐶)𝑔^𝑙𝑑𝜇,(𝐶)𝑔^𝑟𝑑𝜇=(𝐶)𝐺𝑑𝜇. (3.24)
(2) By the same method of (1) and Theorem 2.8(2), we have (𝐶)[𝑎𝐹+𝑏]𝑑𝜇=(𝐶)𝑎𝑓^𝑙+𝑏𝑑𝜇,(𝐶)𝑎𝑓^𝑟=𝑎(𝐶)𝑓+𝑏𝑑𝜇_𝑙𝑑𝜇+𝑏,𝑎(𝐶)𝑓^𝑟𝑑𝜇+𝑏=𝑎(𝐶)𝐹𝑑𝜇+𝑏. (3.25)
(3) The proof is similar to (2).
(4) Since 𝐹∨𝐺=[𝑓^𝑙∨𝑔^𝑙,𝑓^𝑟∨𝑔^𝑟], by Theorem 2.8(4), (𝐶)𝐹∨𝐺𝑑𝜇=(𝐶)𝑓^𝑙∨𝑔^𝑙𝑑𝜇,(𝐶)𝑓^𝑟∨𝑔^𝑟≥𝐶)𝑓𝑑𝜇(^𝑙𝑑𝜇∨(𝐶)𝑔^𝑙𝑑𝜇,(𝐶)𝑓^𝑟𝑑𝜇∨(𝐶)𝑔^𝑟=𝐶)𝑓𝑑𝜇(^𝑙𝑑𝜇,(𝐶)𝑓^𝑟𝑑𝜇∨(𝐶)𝑔^𝑙𝑑𝜇,(𝐶)𝑔^𝑟=(𝐶)(𝐶)𝑑𝜇𝐹𝑑𝜇∨𝐺𝑑𝜇. (3.26)
(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 𝐴∈𝔅 and {𝐹_𝑛} is equally (𝐶) integrable and If {𝐹_𝑛} converges in the (𝐶) to 𝐹 and 𝜇 is autocontinuous, then lim_𝑛→∞𝑑_𝐻(𝐶)_𝐴𝐹_𝑛𝑑𝜇,(𝐶)_𝐴𝐹𝑑𝜇=0. (3.27)

Proof. Since {𝐹_𝑛} is equally (𝐶) integrable on 𝐴, by Theorem 3.7, {𝑓^𝑙_𝑛} and {𝑓^𝑟_𝑛} are equally (𝐶) integrable on 𝐴. By Theorem 2.9, lim_𝑛→∞(𝐶)_𝐴𝑓^𝑙_𝑛𝑑𝜇=(𝐶)_𝐴𝑓^𝑙𝑑𝜇,lim_𝑛→∞(𝐶)_𝐴𝑓^𝑟_𝑛𝑑𝜇=(𝐶)_𝐴𝑓^𝑟𝑑𝜇. (3.28) Thus, for any given 𝜀>0, there exists 𝐾 such that (𝐶)_𝐴𝑓^𝑙_𝑛𝑑𝜇−(𝐶)_𝐴𝑓^𝑙𝐶)𝑑𝜇<𝜀,(_𝐴𝑓^𝑟_𝑛𝑑𝜇−(𝐶)_𝐴𝑓^𝑟𝑑𝜇<𝜀, (3.29) for all 𝑛≥𝐾. Then, we have 𝑑_𝐻(𝐶)_𝐴𝐹_𝑛𝑑𝜇,(𝐶)_𝐴𝐹𝑑𝜇<𝜀, (3.30) for all 𝑛≥𝐾. 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 [2–6, 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.