# On Properties of the Choquet Integral of Interval-Valued Functions

Authors: Lee-Chae Jang
Journal: Journal of Applied Mathematics
 (2011)

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## 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.


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## Body

## 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 Section2, 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⊂⋯⊂⋯,An∈𝔅(n=1,2,…)⇒limn→∞μ(An)=μ(∪n=1∞An) (continuity from below);(FM4)
A1⊃A2⊃⋯⊃An⊃⋯,An∈𝔅(n=1,2,…) and μ(A1)<∞⇒limn→∞μ(An)=μ(∩n=1∞An) (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 measureμ is said to be autocontinuous from above (resp., below) if A∈𝔉, {Bn}⊂𝔉, and limn→∞μ(Bn)=0⇒limn→∞μ(A∪Bn)=μ(A) (resp., limn→∞μ(A∖Bn)=μ(A)).
(2) Ifμ is autocontinuous both from above and from below, it is said to be autocontinuous.Definition 2.3 (see [1, 2, 4, 5]).
(1) LetA∈𝔅 and f∈𝔉.The Choquet integral of f with respect to a fuzzy measure μ is defined by
(2.1)(C)∫Afdμ=∫0∞μ(Gf(α)∩A)dα,
where the integral on the right-hand side is the Lebesgue integral and Gf(α)={x∣f(x)≥α}.
(2) A measurable functionf 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,…,xn} and 𝔅=℘(X) is the power set of X and f is a measurable function on X, then (2.2)(C)∫Afdμ=∑i=1nf(x(i))[μ(A(i))-μ(A(i+1))],
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 (2.3)(C)∫Afdμ=∑i=1n(f(x(i))-f(x(i-1)))μ(A(i)).Definition 2.4 (see [1, 2]).
Let{fn}⊂𝔉 and f∈𝔉. A sequence {fn} converges in the (C) mean to f if
(2.4)lim⁡n→∞(C)∫A|fn-f|dμ=0.Definition 2.5 (see [1, 2]).
LetA∈𝔅. A sequence {fn} is called equally (C) integrable on A if for any given ε>0, there exists N(ε)>0 such that
(2.5)(C)∫Afndμ≤∫0Nμ(Gfn(α)∩A)dα+ε
for all n=1,2,….It is easy to see that if there exists a(C) integrable function g such that |fn|≤g for all n=1,2,…, then {fn} is equally (C) integrable.Definition 2.6 (see [2]).
LetA∈𝔅, {fn}⊂𝔉, and f∈𝔉. We say that {fn} converges in μ to f on A if for any given ε>0,
(2.6)lim⁡n→∞μ({x∣|fn(x)-f(x)|≥ε}∩A)=0.Definition 2.7 (see [2, 4, 5]).
Letf,g∈𝔉. f and g are comonotonic if for every pair x,y∈X,
(2.7)f(x)<f(y)⟹g(x)≤g(y).Theorem 2.8 (see [1–5]).
LetA∈𝔅,  f,g∈𝔉, and  μ be a fuzzy measure.
(1) Iff≤g, then  (C)∫Afdμ≤(C)∫Agdμ.
(2) Ifa and b are nonnegative real numbers, then
(2.8)(C)∫A(af+b)dμ=a(C)∫Afdμ+bμ(A).
(3) Iff and g are comonotonic, then
(2.9)(C)∫A(f+g)dμ=(C)∫Afdμ+(C)∫Agdμ.
(4) If we define(f∨g)(x)=f(x)∨g(x) for all x∈X, then
(2.10)(C)∫Af∨gdμ≥(C)∫Afdμ∨(C)∫Agdμ.
(5) If we define(f∧g)(x)=f(x)∧g(x) for all x∈X, then
(2.11)(C)∫Af∧gdμ≤(C)∫Afdμ∧(C)∫Agdμ.Theorem 2.9 (see [1, 2]).
LetA∈𝔅 and  {fn} be equally  (C) integrable. If  {fn} converges in the  (C) mean to  f and  μ is autocontinuous, then
(2.12)lim⁡n→∞(C)∫Afndμ=(C)∫Afdμ.Note that if{fn} satisfies (2.12), then it is said to be Choquet weak converge to f.


## 3. Interval-Valued Functions and the Choquet Integral

LetC(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, (3.1)I(R)={[al,ar]∣al,ar∈R,al≤ar}.
For any a∈ℝ, we define a=[a,a]. Obviously, a∈I(ℝ) (see [9, 10, 14, 15]).Definition 3.1.
Ifa̅=[al,ar], b̅=[bl,br]∈I(ℝ+), and k∈ℝ+, then we define arithmetic, maximum, minimum, order, and inclusion operations as follows:(1)
a̅+b̅=[al+bl,ar+br],(2)
ka̅=[kal,kar],(3)
a̅b̅=[albl,arbr],(4)
a̅∨b̅=[al∨bl,ar∨br],(5)
a̅∧b̅=[al∧bl,ar∧br],(6)
a̅≤b̅ if and only if al≤bl and ar≤br,(7)
a̅<b̅ if and only if a̅≤b̅ and a̅≠b̅, and(8)
a̅⊂b̅ if and only if bl≤al and ar≤br.Let𝔉c be the set of all measurable closed set-valued functions on (X,𝔅) and 𝔉i 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+, (3.2)F-1(O)={x∈R+∣F(x)∩O≠∅}∈B.
Then, we introduce the Choquet integral of measurable interval-valued functions (see [7–11]).Definition 3.2 (see [11]).
(1) LetA∈𝔅,F∈𝔉c and μ be a fuzzy measure. The Choquet integral of F with respect to μ on A is defined by
(3.3)(C)∫AFdμ={(C)∫Afdμ∣f∈Sc(F)},
where Sc(F) is the family of measurable selections of F, that is,
(3.4)Sc(F)={f:X→R+∣(C)∫Afdμ<∞,f(x)∈F(x)μ-a.e.}.
(2)F is said to be (C) integrable if (C)∫Fdμ≠∅.
(3)F is said to be (C) integrably function f such that
(3.5)‖F(x)‖=sup⁡r∈F(x)r≤g(x)∀x∈X.We note thatμ-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)]).
LetA∈𝔅 and  μ be a fuzzy measure. If  F=[fl,fr]:X→I(R+) is a  (C) integrably bounded interval-valued function, then
(3.6)(C)∫Fdμ=[(C)∫fldμ,(C)∫frdμ].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{Fn}⊂𝔉i and F∈𝔉i. A sequence {Fn} converges in the (C) mean to F if
(3.7)lim⁡n→∞(C)∫dH(Fn(x),F(x))dμ=0,
where dH is the Hausdorrf metric on C(ℝ+).Recall that for each pairA,B∈C(ℝ+), (3.8)dH(A,B)=max⁡{sup⁡y∈Binf⁡x∈A|x-y|,sup⁡x∈Ainf⁡y∈B|x-y|}.
Then, it is easy to see that for each pair a̅=[al,ar], b̅=[bl,br]∈I(ℝ+), (3.9)dH(a̅,b̅)=max⁡{|al-bl|,|ar-br|}.
By (3.7) and (3.9), we obtain the following theorem.Theorem 3.5.
Let{Fn=[fnl,fnr]}⊂𝔉i and  F=[fl,fr]∈𝔉. If a sequence  {Fn} converges in the  (C) mean to  F, then a sequence  {fnl} (resp.,  {fnr}) converges in the  (C) mean to  fl (resp.,  fr).Proof.
By (3.7) and Theorem 2.8 (4), we have
(3.10)(C)∫dH(Fn(x),F(x))dμ(x)=(C)∫max⁡{|fnl(x)-fl(x)|,|fnr(x)-fr(x)|}dμ≥(C)∫|fnl(x)-fl(x)|dμ(x)∨(C)∫|fnr(x)-fr(x)|dμ(x).
From (3.10), we can derive the followings:
(3.11)lim⁡n→∞(C)|fnl-fl|dμ=0,lim⁡n→∞(C)|fnr-fr|dμ=0.
Thus, the proof is complete.Definition 3.6.
LetA∈𝔅. A sequence {Fn} is called equally (C) integrable on A if for any given ε>0, there exists N(ε)>0 such that
(3.12)dH((C)∫AFndμ,[∫0Nμ(Gfnl(α)∩A)dα,∫0Nμ(Gfnr(α)∩A)dα])≤ε,
for all n=1,2,….Theorem 3.7.
LetA∈𝔅 and  μ be a fuzzy measure. If a sequence  {Fn} is equally  (C) integrable on  A, then sequences  {fnl} and  {fnr} are equally  (C) integrable on  A.Proof.
By (3.9) and (3.12), we have
(3.13)dH((C)∫AFndμ,[∫0Nμ(Gfnl(α)∩A)dα,∫0Nμ(Gfnr(α)∩A)dα])=max⁡{|(C)∫Afldμ-∫0Nμ(Gfnl(α)∩A)dα|,|(C)∫Afrdμ-∫0Nμ(Gfnr(α)∩A)dα|}.
By the assumption and (3.13), given ε>0, there exists N(ε)>0 such that
(3.14)max⁡{|(C)∫Afldμ-∫0Nμ(Gfnl(α)∩A)dα|,|(C)∫Afrdμ-∫0Nμ(Gfnr(α)∩A)dα|}≤ε.
Thus we have
(3.15)(C)∫Afldμ-∫0Nμ(Gfnl(α)∩A)dα≤ε,(C)∫Afrdμ-∫0Nμ(Gfnr(α)∩A)dα≤ε.
Thus, the proof is complete.Definition 3.8.
LetA∈𝔅,{Fn}⊂𝔉i, and F∈𝔉i. We say that {Fn} converges in μ to F on A if for any given ε>0,
(3.16)lim⁡n→∞μ({x∣dH(Fn(x),F(x))≥ε}∩A)=0.Theorem 3.9.
LetA∈𝔅,{Fn=[fnl,fnr]}⊂𝔉i, and  F=[fl,fr]∈𝔉i. If  {Fn} converges in  μ to  F on  A, then a sequence  {fnl} (resp.,  {fnr}) converges in  μ to  fl (resp.,  fr) on  A.Proof.
By (3.9), we have
(3.17)dH(Fn(x),F(x))=max⁡{|fnl(x)-fl(x)|,|fnr(x)-fr(x)|},∀x∈X.
Since  {x∣|fnl(x)-fl(x)|≥ε}⊂{x∣dH(Fn(x),F(x))≥ε} and {x|fnr(x)-fr(x)|≥ε}⊂{x∣dH(Fn(x),F(x))≥ε}, by (3.16), we have
(3.18)lim⁡n→∞μ({x∣|fnl(x)-fl(x)|≥ε}∩A)=0,lim⁡n→∞μ({x∣|fnr(x)-fr(x)|≥ε}∩A)=0.
Thus, the proof is complete.Definition 3.10.
LetF,G∈𝔉i. F and G are comonotonic if and only if for every pair x,y∈X,(3.19)F(x)<F(y)⟹F(x)≤G(y).Theorem 3.11.
LetF=[fl,fr], G=[gl,gr]∈𝔉i. If  F and  G are comonotonic, then  fl and  gl are comonotonic, and  fr and  gr are comonotonic.Proof.
Ifx,y∈X and fl(x)<fl(y) and fr(x)<fr(y), then we have F(x)<G(y). Since F and G are comonotonic, G(x)≤G(y). Then, we have gl(x)≤gl(y) and  gr(x)≤gr(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.
LetA∈𝔅,  F,G∈𝔉i, and let  μ be a fuzzy measure.
(1) IfF≤G, then  (C)∫AFdμ≤(C)∫AGdμ.
(2) Ifa and b are nonnegative real numbers, then
(3.20)(C)∫A∫(aF+b)dμ=a(C)∫AFdμ+bμ(A).
(3) IfF and G are comonotonic, then
(3.21)(C)∫A(F+G)dμ=(C)∫AFdμ+(C)∫AGdμ.
(4) If we define(F∨G)(x)=F(x)∨G(x) for all x∈X, then
(3.22)(C)∫AF∨Gdμ≥(C)∫AFdμ∨(C)∫AGdμ.
(5) If we define(F∧G)(x)=F(x)∧G(x) for all x∈X, then
(3.23)(C)∫AF∧Gdμ≤(C)∫AFdμ∧(C)∫AGdμ.Proof.
(1) IfF=[fl,fr]≤G=[gl,gr], then fl≤gl and fr≤gr. By Theorem 2.8(1), (C)∫fldμ≤(C)∫gldμ and (C)∫frdμ≤(C)∫grdμ. By Definition 3.1(6) and (3.6), we have
(3.24)(C)∫Fdμ=[(C)∫fldμ,(C)∫frdμ]≤[(C)∫gldμ,(C)∫grdμ]=(C)∫Gdμ.
(2) By the same method of (1) and Theorem2.8(2), we have
(3.25)(C)∫[aF+b]dμ=[(C)∫afl+bdμ,(C)∫afr+bdμ]=[a(C)∫fldμ+b,a(C)∫frdμ+b]=a(C)∫Fdμ+b.
(3) The proof is similar to (2).
(4) SinceF∨G=[fl∨gl,fr∨gr], by Theorem 2.8(4),
(3.26)(C)∫F∨Gdμ=[(C)∫fl∨gldμ,(C)∫fr∨grdμ]≥[(C)∫fldμ∨(C)∫gldμ,(C)∫frdμ∨(C)∫grdμ]=[(C)∫fldμ,(C)∫frdμ]∨(C)[∫gldμ,(C)∫grdμ]=(C)∫Fdμ∨(C)∫Gdμ.
(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.
IfA∈𝔅 and  {Fn} is equally  (C) integrable and If  {Fn} converges in the  (C) to  F and  μ is autocontinuous, then
(3.27)lim⁡n→∞dH((C)∫AFndμ,(C)∫AFdμ)=0.Proof.
Since{Fn} is equally (C) integrable on A, by Theorem 3.7, {fnl} and {fnr} are equally (C) integrable on A. By Theorem 2.9,
(3.28)lim⁡n→∞(C)∫Afnldμ=(C)∫Afldμ,lim⁡n→∞(C)∫Afnrdμ=(C)∫Afrdμ.
Thus, for any given ε>0, there exists K such that
(3.29)|(C)∫Afnldμ-(C)∫Afldμ|<ε,|(C)∫Afnrdμ-(C)∫Afrdμ|<ε,
for all n≥K. Then, we have
(3.30)dH((C)∫AFndμ,(C)∫AFdμ)<ε,
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.



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