Intervals can be used in the representation of uncertainty. In this regard, we consider monotone intervalvalued set functions and the Choquet integral. This paper investigates characterizations of monotone intervalvalued set functions and provides applications of the Choquet integral with respect to monotone intervalvalued set functions, on the space of measurable functions with the Hausdorff metric.
1. Introduction
Axiomatic characterizations of the Choquet integral have been introduced by Choquet
[1]
, Murofushi et al
[2
,
3]
, Wang
[4]
and CamposBolanos
[5]
as an interesting extension of the Lebesgue integral. Other researchers have studied various convergence problems on monotone set functions and, on sequences of measurable functions, as well as applications. For example, the convergence in the (C) mean
[6]
, on decisionmaking problems
[7
,
8]
, on the Choquet weak convergence
[9]
, on the monotone expectation
[10]
, and on the aggregation approach
[11]
.
In the past decade, it has been suggested to use intervals in order to represent uncertainty, for example, for economic uncertainty
[12]
, for fuzzy random variables
[13]
, in intervalprobability
[14]
, for martingales of multivalued functions
[15]
, in the integrals of setvalued functions
[16]
, in the Choquet integrals of intervalvalued (or closed setvalued) functions
[17

22]
, and for intervalvalued capacity functions
[23]
. CousoMontesGil
[24]
studied applications under the sufficient and necessary conditions on monotone set functions, i.e., the subadditivity of the Choquet integral with respect to monotone set functions.
Intervals are useful in the representation of uncertainty. We shall consider monotone intervalvalued set functions and the Choquet integral with respect to a monotone intervalvalued set function of measurable functions. Based on the results of CousoMotesGil
[24]
, we shall provide characterizations of monotone intervalvalued set functions as well as applications of the Choquet integral regarding a monotone intervalvalued set function in the space of measurable functions with the Hausdorff metric.
In Section 2, we list definitions and basic properties for the monotone set functions, the Choquet integrals and for the various convergence notions in the space of measurable functions.
In Section 3, we define a monotone intervalvalued set function and the Choquet integral with respect to a monotone intervalvalued set function of measurable functions, and we discuss their properties. We also investigate various convergences in the Hausdorff metric on the space of intervals as well as the characterizations of the Choquet integral with respect to a monotone intervalvalued set function of measurable functions.
In Section 4, we give a brief summary of our results and conclusions.
2. Preliminaries and Definitions
In this section, we consider monotone set functions, also called fuzzy measures, and the Choquet integral defined by Choquet
[1]
. The Choquet integral
[1]
generalizes the Lebesgue integral to the case of monotone set functions. Let
X
be a nonempty set, and let
A
denote a
σ
algebra of subsets of
X
. Let
and
First we define, monotone set functions, the Choquet integral, the different types of convergences, and the uniform integrability of measurable functions as follows:
Definition 2..1.
[2
,
3
,
5
,
24]
(1) A mapping
is said to be a set function if
μ
(Ø)=0.
(2) A set function
μ
is said to be monotone if
(3) A set function
μ
is said to be continuous from below (or lower semicontinuous) if for any sequence {
A_{n}
} ⊂
A
and
A
∈
A
such that
(4) A set function
μ
is said to be continuous from above (or upper semicontinuous) if for any sequence {
A_{n}
} ⊂
A
and
A
∈
A
such that
(5) A set function
μ
is said to be continuous if it is continuous from above and continuous from below.
(6) A set function
μ
is said to be subadditive if
A
,
B
∈
A
and A ∩ B = Ø, then
(7) A set function
μ
is said to be submodular if
A
,
B
∈
A
, then
(8) A set function
μ
is said to be nulladditive if
Definition 2..2.
[2
,
3
,
5
,
24]
Let
μ
be a monotone set function on
A
. (1) If
f
:
X
→ ℝ
^{+}
is a nonnegative measurable function, then the Choquet integral of
f
with respect to
μ
is defined by
where
for all
α
∈ 2 ℝ
^{+}
and the integral on the righthand side is the Lebesgue integral of
μ_{f}
.
(2) If
f
:
X
→ ℝ is a realvalued measurable function, then the Choquet integral of
f
with respect to
μ
is defined by
where
f
^{+}
= max{
f
,0},
f
^{–}
=max{–
f
,0},
A^{c}
is the complementary set of
A
, and
μ
^{*}
is the conjugate of
μ
, that is,
(3) A measurable function
f
is said to be
μ
integrable if the Choquet integral of
f
on
X
exists.
We note that
for all
α
∈ ℝ
^{+}
and
for all
α
∈ ℝ
^{–}
= (–∞, 0). Thus, we have
We introduce almost everywhere convergence, convergence in
μ
mean, and uniform
μ
integrability as follows:
Definition 2..3.
Let
μ
be a monotone set function on a measurable space (
X
,
A
), {
f_{n}
} a sequence of measurable functions from
X
to ℝ, and
f
a measurable function from
X
to ℝ.
(1) A sequence {
f_{n}
} almost everywhere converges to
f
if there exists a measurable and null subset
N
∈
A
,
μ
(
N
) = 0 such that
(2) A sequence {
f_{n}
} converges in
μ
mean to
f
if
where  ᠊  is the absolute value on ℝ.
Definition 2..4.
[24]
Let
μ
be a monotone set function on
A
and
I
⊂ ℕ an index set. A class of realvalued measurable functions {
f_{n}
}
_{n∈I}
is said to be uniform
μ
integrable if
Now, we recall from
[24]
the subadditivity of the Choquet integral, the equivalence between the convergence in mean and the uniform integrability of a sequence of measurable functions.
Theorem 2..5.
(Subadditivity for the Choquet integral) Let (
X
,
A
) be a measurable space. If a monotone set function
μ
:
A
→ ℝ
^{+}
is submodular and
f
,
g
:
X
→ ℝ are realvalued measurable functions, then we have
Theorem 2..6.
Let (
X
,
A
) be a measurable space. If a monotone set function
μ
:
A
→ ℝ
^{+}
is subadditive and
f
,
g
:
X
→ ℝ are measurable functions with disjoint support, that is, {
x
∈
X

f
(
x
) > 0}∩{
x
∈
X

g
(
x
) > 0} = Ø, then we have
3. Main Results
In this section, we consider intervals, intervalvalued functions, and the Aumann integral of measurable intervalvalued functions. Let
I
(ℝ) be the class of all bounded and closed intervals (intervals, for short) in ℝ as follows:
For any
a
∈ ℝ, we define
a
= [
a
,
a
]. Obviously,
a
∈
I
(ℝ)
[18

21]
.
Recall that if (ℝ,
ᙏ
,
m
) is the Lebesgue measure space and
C
(ℝ) is the set of all closed subsets of ℝ, then the Aumann integral of a closed setvalued function
F
: ℝ →
C
(ℝ) is defined by
where
S
(
F
) is the set of all integrable selections of
F
, that is,
where
m
–
a.e.
means almost everywhere in the Lebesgue measure
m
, and 
g
 is the absolute value of
g
[15
,
16]
. In
[13
,
23]
, we can see that (
A
)
ʃ Fdm
is a nonempty bounded and closed interval in ℝ whenever
F
is an intervalvalued function as in the following theorem.
Theorem 3..1.
If an intervalvalued function
F
= [
g_{l}
,
g_{r}
] : ℝ →
I
(ℝ) is measurable and integrably bounded, then
g_{l}
,
g_{r}
∈
S
(
F
) and
where the two integrals on the righthand side are the Lebesgue integral with respect to
m
.
Note that we write
for all bounded continuous function
g
. Let
C
(ℝ) be the class of all closed ℝ. We recall that the Hausdorff metric
d_{H}
:
C
(ℝ) ×
C
(ℝ) → ℝ
^{+}
is defined by
for all
A
,
B
∈
C
(ℝ). It is wellknown that for all
ā
= [
a_{l}
,
a_{r}
],
Next, we shall define monotone intervalvalued set functions and discuss their characterization.
Definition 3..2.
(1) A mapping
is said to be an intervalvalued set function if
(2) An intervalvalued set function
is said to be monotone if
(3) An intervalvalued set function
is said to be continuous from below if for any sequence {
A_{n}
}⊂
A
and
A
∈
A
such that
A_{n}
↑
A
, then
that is,
(4) An intervalvalued set function
is said to be continuous from above if for any sequence {
A_{n}
}⊂
A
and
A
∈
A
such that
is a bounded interval and
A_{n}
↓
A
, then
(5) An intervalvalued set function
is said to be continuous if it is both continuous from above and continuous from below.
(6) An intervalvalued set function
is said to be subadditive if
A
,
B
∈
A
, then
(7) An intervalvalued set function
is said to be submodular if
A
,
B
∈
A
, then
(8) An intervalvalued set function
is said to be nulladditive if
From Definition 3.2 and Eq. (25), we can directly derive the following theorem
[23
,
25]
.
Theorem 3..3.
(1) A mapping
is an intervalvalued set function if only only if
μ_{l}
and
μ_{r}
are set functions, and
μ_{l}
≤
μ_{r}
.
(2) An intervalvalued set function
is monotone if only only if the set functions
μ_{l}
and
μ_{r}
are monotone.
(3) An intervalvalued set function
is continuous from below if only only if the set functions
μ_{l}
and
μ_{r}
are continuous from below, and
μ_{l}
≤
μ_{r}
.
(4) An intervalvalued set function
is continuous from above if only only if the set functions
μ_{l}
and
μ_{r}
are continuous from above, and
μ_{l}
≤
μ_{r}
.
(5) An intervalvalued set function
is subadditive if and only if the set functions
μ_{l}
and
μ_{r}
are subadditive, and
μ_{l}
≤
μ_{r}
.
(6) An intervalvalued set function
is submodular if and only if the set functions
μ_{l}
and
μ_{r}
are submodular, and
μ_{l}
≤
μ_{r}
.
(7) An intervalvalued set function
is nulladditive if and only if the set functions
μ_{l}
and
μ_{r}
are nulladditive, and
μ_{l}
≤
μ_{r}
.
By using Definition 2.2 and Theorem 3.3, we define the Choquet integral of a nonnegative measurable function with respect to a continuous from below and monotone intervalvalued set function as follows:
Definition 3..4.
(1) The Choquet integral of a nonnegative measurable function
f
:
X
→ ℝ
^{+}
, with respect to a monotone intervalvalued set function
is defined by
where
m
is the Lebesgue measure on ℝ and the integral on the righthand side is the Aumann integral with respect to
m
of
(2) The Choquet integral of a realvalued measurable function
f
:
X
→ ℝ, with respect to a monotone intervalvalued set function
is defined by
where
f
^{+}
= max{
f
,0} and
f
^{–}
=max{–
f
,0}, and
is the conjugate of
that is,
(3) A measurable function
f
is said to be
integrable if
We note that Eq. (36) implies
where
for all
α
∈ ℝ
^{+}
. By the definition of
we easily get the following theorem.
Theorem 3..5.
(1) A monotone intervalvalued set function
is continuous from below (resp. from above) if and only if
is continuous from above (resp. from below).
(2) If
is a monotone intervalvalued set function and
μ_{l}
(
X
) =
μ_{r}
(
X
), then
where
In
[21]
, we can find the theorem below. This gives a useful and interesting tool for the application of the Choquet integral of a nonnegative measurable function
f
, with respect to a monotone intervalvalued set function
Theorem 3..6.
([23, Lemma 2.5 (i) and (v)]) Let
f
be a nonnegative measurable function and
a monotone intervalvalued function. If
is continuous from above and we take
for all
α
∈ ℝ
^{+}
, then we have
(1)
F
is continuous from above, and
(2)
where
m
is the Lebesgue measure and
μ_{lf}
(
α
)=
μ_{l}
({
x
∈
X

f
(
x
) >
α
}) and
μ_{rf}
(
α
)=
μ_{r}
({
x
∈
X

f
(
x
) >
α
}).
Note that Theorem 3.6(2) implies the following equation (36) under the same condition of
f
and
By using Theorem 3.5 and Eq. (36), we can obtain the following theorem, which is a useful formula for the Choquet integral of a measurable function
f
:
X
→ ℝ, with respect to a continuous monotone intervalvalued set function.
Theorem 3..7.
Let
f
be a measurable function and
a monotone intervalvalued set function. If
is continuous and
μ_{l}
(
X
) =
μ_{r}
(
X
), then we have
Proof.
Let
f
^{+}
= max{
f
, 0} and
f
^{–}
= max{–
f
, 0}. Since
is continuous from above, by (40), we have
Since
is continuous from below and
μ_{l}
(
X
) =
μ_{r}
(
X
), by Theorem 3.5(2),
is continuous from above. Thus, by (36), we have
By Definition 3.4(2), Eq. (38), and Eq. (39), we have the result.
Next, we present the following theorems which give characterizations of the Choquet integral with respect to a monotone intervalvalued set function.
Theorem 3..8.
Let
be a monotone intervalvalued set function and let
A
∈
A
. If
is continuous from above, then we have
Proof.
If
a
≥ 0, then by Eq. (36), we have
If
a
< 0, then by Eq. (36), we have
Theorem 3..9.
Let a monotone intervalvalued set function
be continuous from above and let
f
a nonnegative
integrable function. If
is continuous from above and
A
,
B
∈
A
with
A
⊂
B
, then we have
Proof.
Since
is a monotone intervalvalued set function, by Theorem 3.3 (1) and (2),
μ_{l}
and
μ_{r}
are monotone intervalvalued set functions. Thus,
By Eq. (36) and Eq. (42), we have the result.
We remark that if we take a
integrable function
f
which is
f
^{+}
= 0 and
f
^{–}
> 0, then
is not monotone, that is, for each pair
A
,
B
⊂
A
with
A
∈
B
,
Theorem 3..10.
Let
be a monotone intervalvalued set function which is continuous from above, and let
A
∈
A
. If
f
and
g
are nonnegative
integrable functions with
f
≤
g
, then we have
Proof.
The proof is similar to the proof of Theorem 3.10.
Theorem 3..11.
Let
be monotone intervalvalued set functions,
f
a nonnegative
integrable and
integrable function, and
A
∈
A
.
(1) If
then we have
(2) If
then we have
Proof.
(1) Note that
if and only if
μ
_{1l}
≤
μ
_{2l}
and
μ
_{lr}
≤
μ
_{2r}
. Thus, we have
By (36) and (47), we have the result.
(2) Note that
if and only if
μ
_{2l}
≤
μ
_{1l}
and
μ
_{1r}
≤
μ
_{2r}
. Thus, we have
By Eq. (36) and eq. (48), we have the result.
Finally, we investigate the subadditivity of the Choquet integral under some conditions for the monotone intervalvalued set functions.
Theorem 3..12.
Let (
X
,
A
) be a measurable space. If a continuous monotone intervalvalued set function
with
μ_{l}
(
X
) =
μ_{r}
(
X
), is submodular and
are measurable functions, then we have
Proof.
Since
is a submodular monotone intervalvalued set function, by Theorem 3.3(6),
μ_{l}
and
μ_{r}
are submodular monotone set functions.
By Theorem 2.5, we have
and
By Eq. (36), eq. (50), and eq. (51), we have the result.
Theorem 3..13.
Let (
X
,
A
) be a measurable space. If a continuous monotone intervalvalued set function
with
μ_{l}
(
X
) =
μ_{r}
(
X
), is subadditive, and
are measurable functions with disjoint support, then
Proof.
Since
is a subadditive monotone intervalvalued set function, by Theorem 3.3(5),
μ_{l}
and
μ_{r}
are subadditive monotone set functions. By Theorem 2.6, we have
and
By Eq. (36), Eq. (53), and Eq. (54), we have the result.
4. Conclusions
In this paper, we introduced the concept of a monotone intervalvalued set function and, the Aumann integral of a measurable function, with respect to the Lebesgue measure. By using the two concepts, we define the Choquet integral with a monotone intervalvalued set function of measurable functions.
From Theorem 3.2, Definition 3.3(3), and the condition that
μ_{l}
(
X
) =
μ_{r}
(
X
) of a continuous monotone set function, we can deal with the new concept of the Choquet integral of a monotone intervalvalued set function
of measurable functions
f
:
X
→ ℝ. Theorems 3.3, 3.5, 3.6, 3.7, 3.8, 3.9, and 3.10 are important characterizations of the Choquet integral with respect to a monotone intervalvalued set function on the space of nonnegative
integrable functions. Theorem 3.12 and Theorem 3.13 are both, useful and interesting tools, in the application of the Choquet integral with respect to a continuous monotone interval valued set function.
In the future, by using the results in this paper, we shall investigate various problems and models, for representing monotone uncertain set functions, and for the application of the biChoquet integral with respect to a monotone intervalvalued set function
 Conflict of Interest
No potential conflict of interest relevant to this article was reported.
Acknowledgements
This work was supported by Konkuk University in 2013.
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