Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 492149, 10 pages
doi:10.1155/2011/492149
Research Article
On Properties of the Choquet Integral of
Interval-Valued Functions
Lee-Chae Jang
Department of Computer Engineering, Konkuk University, Chungju 138-701, Republic of Korea
Correspondence should be addressed to Lee-Chae Jang, leechae.jang@kku.ac.kr
Received 16 July 2011; Accepted 6 September 2011
Academic Editor: F. Marcellán
Copyright q 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.
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 intervalvalued 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
2
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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 σalgebra 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 ⇒ μA ≤ μB monotonicity;
FM3 A1 ⊂ A2 ⊂ · · · ⊂ · · · , An ∈ B n 1, 2, . . . ⇒ limn → ∞ μAn μ∪∞
n1 An continuity from below;
FM4 A1 ⊃ A2 ⊃ · · · ⊃ An ⊃ · · · , An ∈ B n 1, 2, . . . and μA1 < ∞ ⇒ limn → ∞ μAn μ∩∞
n1 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 measure μ is said to be autocontinuous from above resp.,
below if A ∈ F, {Bn } ⊂ F, and limn → ∞ μBn 0 ⇒ limn → ∞ μA ∪ Bn μA resp.,
limn → ∞ μA \ Bn μA.
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
fdμ C
∞
μ Gf α ∩ A dα,
2.1
0
A
where the integral on the right-hand side is the Lebesgue integral and Gf α {x | fx ≥ 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 X fdμ, 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
fdμ C
A
n
f xi μ Ai − μ Ai1 ,
i1
2.2
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3
where · is a permutation on {1, 2, . . . , n} such that 0 fx0 ≤ fx1 ≤ · · · ≤ fxn and
Ai {xi , xi1 , . . . , xn } and An1 ∅. From 2.2, clearly we have
fdμ C
n
f xi − f xi−1 μ Ai .
A
2.3
i1
Definition 2.4 see 1, 2. Let {fn } ⊂ F and f ∈ F. A sequence {fn } converges in the C mean
to f if
fn − f dμ 0.
lim C
n→∞
2.4
A
Definition 2.5 see 1, 2. Let A ∈ B. A sequence {fn } is called equally C integrable on A if
for any given ε > 0, there exists Nε > 0 such that
fn dμ ≤
C
N
μ Gfn α ∩ A dα ε
2.5
0
A
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. Let A ∈ B, {fn } ⊂ F, and f ∈ F. We say that {fn } converges in μ to f
on A if for any given ε > 0,
lim μ
n→∞
x | fn x − fx ≥ 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,
fx < f y ⇒ gx ≤ g y .
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 A fdμ ≤ C A gdμ.
(2) If a and b are nonnegative real numbers, then
af b dμ aC
C
A
fdμ bμA.
2.8
A
(3) If f and g are comonotonic, then
fdμ C
f g dμ C
C
A
A
gdμ.
A
2.9
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(4) If we define f ∨ gx fx ∨ gx for all x ∈ X, then
f ∨ gdμ ≥ C
C
fdμ ∨ C
A
A
gdμ.
2.10
gdμ.
2.11
A
(5) If we define f ∧ gx fx ∧ gx for all x ∈ X, then
f ∧ gdμ ≤ C
C
A
fdμ ∧ C
A
A
Theorem 2.9 see 1, 2. Let A ∈ B and {fn } be equally C integrable. If {fn } converges in the
C mean to f and μ is autocontinuous, then
fn dμ C
lim C
n→∞
A
fdμ.
2.12
A
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
Let CR be the set of all closed subsets in R and IR the set of all bounded closed intervals
intervals, for short in R −∞, ∞, that is,
IR al , ar | al , ar ∈ R, al ≤ ar .
3.1
For any a ∈ R, we define a a, a. Obviously, a ∈ IR see 9, 10, 14, 15.
Definition 3.1. If a al , ar , b bl , br ∈ IR , and k ∈ R , then we define arithmetic,
maximum, minimum, order, and inclusion operations as follows:
1 a b al bl , ar br ,
2 ka kal , kar ,
3 ab al bl , ar br ,
4 a ∨ b al ∨ bl , ar ∨ br ,
5 a ∧ b al ∧ bl , ar ∧ br ,
6 a ≤ b if and only if al ≤ bl and ar ≤ br ,
b, and
7 a < b if and only if a ≤ b and a /
8 a ⊂ b if and only if bl ≤ al and ar ≤ br .
Let Fc 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 →
CR is said to be measurable if for any open set O ⊂ R ,
F −1 O {x ∈ R | Fx ∩ O /
∅} ∈ B.
3.2
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5
Then, we introduce the Choquet integral of measurable interval-valued functions see 7–
11.
Definition 3.2 see 11. 1 Let A ∈ B, F ∈ Fc and μ be a fuzzy measure. The Choquet
integral of F with respect to μ on A is defined by
Fdμ C
fdμ | f ∈ Sc F ,
C
A
3.3
A
where Sc F is the family of measurable selections of F, that is,
Sc F f : X → R | C
fdμ < ∞, fx ∈ Fx μ − a.e. .
3.4
A
2 F is said to be C integrable if C Fdμ / ∅.
3 F is said to be C integrably function f such that
Fx sup r ≤ gx
∀ x ∈ X.
3.5
r∈Fx
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 intervalvalued functions.
Theorem 3.3 see 11, Theorem 3.16iii. Let A ∈ B and μ be a fuzzy measure. If F f l , f r :
X → IR is a C integrably bounded interval-valued function, then
C
Fdμ C
f dμ, C
l
f dμ .
r
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 {Fn } ⊂ Fi and F ∈ Fi . A sequence {Fn } converges in the C mean to F if
dH Fn x, Fxdμ 0,
lim C
n→∞
3.7
where dH is the Hausdorrf metric on CR .
Recall that for each pair A, B ∈ CR ,
dH A, B max sup inf x − y , sup inf x − y .
y∈B x∈A
x∈A y∈B
3.8
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Journal of Applied Mathematics
Then, it is easy to see that for each pair a al , ar , b bl , br ∈ IR ,
dH a, b max al − bl , |ar − br | .
3.9
By 3.7 and 3.9, we obtain the following theorem.
Theorem 3.5. Let {Fn fnl , fnr } ⊂ Fi and F f l , f r ∈ F. If a sequence {Fn } converges in the
C mean to F, then a sequence {fnl } (resp., {fnr }) converges in the C mean to f l (resp., f r ).
Proof. By 3.7 and Theorem 2.8 4, we have
C
dH Fn x, Fxdμx
C
max fnl x − f l x, fnr x − f r x dμ
3.10
l
l
≥ C fn x − f xdμx ∨ C fnr x − f r xdμx.
From 3.10, we can derive the followings:
lim Cfnl − f l dμ 0,
lim Cfnr − f r dμ 0.
n→∞
n→∞
3.11
Thus, the proof is complete.
Definition 3.6. Let A ∈ B. A sequence {Fn } is called equally C integrable on A if for any
given ε > 0, there exists Nε > 0 such that
C
dH
A
N
N
Fn dμ,
μ Gfnl α ∩ A dα,
μ Gfnr α ∩ A dα
≤ ε,
0
3.12
0
for all n 1, 2, . . ..
Theorem 3.7. Let A ∈ 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
dH
C
A
N
N
Fn dμ,
μ Gfnl α ∩ A dα,
μ Gfnr α ∩ A dα
0
0
N f l dμ −
μ Gfnl α ∩ A dα,
max C
0
A
N
f r dμ −
μ Gfnr α ∩ A dα .
C
0
A
3.13
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7
By the assumption and 3.13, given ε > 0, there exists Nε > 0 such that
N N
l
r
max C
f dμ −
μ Gfnl α ∩ A dα, C
f dμ −
μ Gfnr α ∩ A dα ≤ ε.
0
0
A
A
3.14
Thus we have
f l dμ −
C
0
A
f r dμ −
C
N μ Gfnl α ∩ A dα ≤ ε,
A
N
0
3.15
μ Gfnr α ∩ A dα ≤ ε.
Thus, the proof is complete.
Definition 3.8. Let A ∈ B, {Fn } ⊂ Fi , and F ∈ Fi . We say that {Fn } converges in μ to F on A if
for any given ε > 0,
lim μ{x | dH Fn x, Fx ≥ ε} ∩ A 0.
3.16
n→∞
Theorem 3.9. Let A ∈ B, {Fn fnl , fnr } ⊂ Fi , and F f l , f r ∈ Fi . If {Fn } converges in μ to F
on A, then a sequence {fnl } (resp., {fnr }) converges in μ to f l (resp., f r ) on A.
Proof. By 3.9, we have
dH Fn x, Fx max fnl x − f l x, fnr x − f r x ,
∀x ∈ X.
3.17
Since {x | |fnl x − f l x| ≥ ε} ⊂ {x | dH Fn x, Fx ≥ ε} and {x|fnr x − f r x| ≥ ε} ⊂ {x |
dH Fn x, Fx ≥ ε}, by 3.16, we have
lim μ
n→∞
lim μ
n→∞
x | fnl x − f l x ≥ ε ∩ A 0,
r
x | fn x − f r x ≥ ε ∩ A 0.
3.18
Thus, the proof is complete.
Definition 3.10. Let F, G ∈ Fi . F and G are comonotonic if and only if for every pair x, y ∈ X,
Fx < F y ⇒ Fx ≤ G y .
3.19
Theorem 3.11. Let F f l , f r , G g l , g r ∈ Fi . If F and G are comonotonic, then f l and g l are
comonotonic, and f r and g r are comonotonic.
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Proof. If x, y ∈ X and f l x < f l y and f r x < f r y, then we have Fx < Gy. Since F
and G are comonotonic, Gx ≤ Gy. 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 ∈ Fi , and let
μ be a fuzzy measure.
(1) If F ≤ G, then C A Fdμ ≤ C A Gdμ.
(2) If a and b are nonnegative real numbers, then
aF bdμ aC
C
A
Fdμ bμA.
3.20
A
(3) If F and G are comonotonic, then
Fdμ C
F Gdμ C
C
A
3.21
Gdμ.
A
A
(4) If we define F ∨ Gx Fx ∨ Gx for all x ∈ X, then
F ∨ Gdμ ≥ C
C
Fdμ ∨ C
A
3.22
Gdμ.
A
A
(5) If we define F ∧ Gx Fx ∧ Gx for all x ∈ X, then
F ∧ Gdμ ≤ C
C
Fdμ ∧ C
A
A
3.23
Gdμ.
A
l
l
r
r
Proof.
If F f l , f r ≤ G g l , g r , then
r f ≤ g and f ≤ g . By Theorem 2.81,
1
l
l
r
C f dμ ≤ C g dμ and C f dμ ≤ C g dμ. By Definition 3.16 and 3.6, we have
C
Fdμ C
f dμ ≤ C
f dμ, C
l
g dμ C
g dμ, C
r
l
r
Gdμ.
3.24
2 By the same method of 1 and Theorem 2.82, we have
C aF bdμ C
aC
af bdμ, C
f dμ b, aC
l
3 The proof is similar to 2.
af bdμ
l
r
f dμ b aC
r
3.25
Fdμ b.
Journal of Applied Mathematics
9
4 Since F ∨ G f l ∨ g l , f r ∨ g r , by Theorem 2.84,
C
F ∨ Gdμ
C f l ∨ g l dμ, C f r ∨ g r dμ
≥ C
f dμ ∨ C
C
f dμ ∨ C
l
r
f dμ ∨ C
f dμ, C
l
C
g dμ, C
l
g dμ, C
r
g r dμ
l
3.26
r
g dμ
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 intervalvalued function.
Theorem 3.13. If A ∈ B and {Fn } is equally C integrable and If {Fn } converges in the C to F
and μ is autocontinuous, then
Fn dμ, C
Fdμ 0.
lim dH C
n→∞
A
3.27
A
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,
lim C
n→∞
A
lim C
n→∞
A
fnl dμ C
f l dμ,
fnr dμ
A
C
3.28
r
f dμ.
A
Thus, for any given ε > 0, there exists K such that
l
l
C
fn dμ − C
f dμ < ε,
A
A
A
A
r
r
C
fn dμ − C
f dμ < ε,
3.29
for all n ≥ K. Then, we have
dH C
Fn dμ, C
Fdμ < ε,
A
for all n ≥ K. Thus, the proof is complete.
A
3.30
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Journal of Applied Mathematics
4. Conclusions
In this paper, we consider the new concept of the Choquet integral of a measurable intervalvalued 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 intervalvalued 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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