Axiomatic characterizations of the Choquet integral have been introduced by Choquet [1], Murofushi et al [2,3], Wang [4] and Campos-Bolanos [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 decision-making 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 multi-valued functions [15], in the integrals of set-valued functions [16], in the Choquet integrals of interval-valued (or closed set-valued) functions [17-22], and for interval-valued capacity functions [23]. Couso-Montes-Gil [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 interval-valued set function of measurable functions. Based on the results of Couso-Motes-Gil [24], we shall provide characterizations of monotone interval-valued set functions as well as applications of the Choquet integral regarding a monotone interval-valued 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 interval-valued set function and the Choquet integral with respect to a monotone interval-valued 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 interval-valued set function of measurable functions.
In Section 4, we give a brief summary of our results and conclusions.
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
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
(2) A set function
(3) A set function
(4) A set function
(5) A set function
(6) A set function
(7) A set function
(8) A set function
Definition 2..2. [2,3,5,24] Let
where
for all
(2) If
where
(3) A measurable function
We note that
for all
for all
We introduce almost everywhere convergence, convergence in
Definition 2..3. Let
(1) A sequence {
(2) A sequence {
where | ？ | is the absolute value on ？.
Definition 2..4. [24] Let
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 (
Theorem 2..6. Let (
In this section, we consider intervals, interval-valued functions, and the Aumann integral of measurable interval-valued functions. Let
For any
Recall that if (？,
where
where
Theorem 3..1. If an interval-valued function
where the two integrals on the right-hand side are the Lebesgue integral with respect to
Note that we write
for all bounded continuous function
for all
Next, we shall define monotone interval-valued set functions and discuss their characterization.
Definition 3..2. (1) A mapping
is said to be an interval-valued set function if
(2) An interval-valued set function
is said to be monotone if
(3) An interval-valued set function
is said to be continuous from below if for any sequence {
that is,
(4) An interval-valued set function
is said to be continuous from above if for any sequence {
is a bounded interval and
(5) An interval-valued set function
is said to be continuous if it is both continuous from above and continuous from below.
(6) An interval-valued set function
is said to be subadditive if
(7) An interval-valued set function
is said to be submodular if
(8) An interval-valued 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
(2) An interval-valued set function
is monotone if only only if the set functions
(3) An interval-valued set function
is continuous from below if only only if the set functions
(4) An interval-valued set function
is continuous from above if only only if the set functions
(5) An interval-valued set function
is subadditive if and only if the set functions
(6) An interval-valued set function
is submodular if and only if the set functions
(7) An interval-valued set function
is null-additive if and only if the set functions
By using Definition 2.2 and Theorem 3.3, we define the Choquet integral of a non-negative 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 non-negative measurable function
is defined by
where
(2) The Choquet integral of a real-valued measurable function
is defined by
where
is the conjugate of
that is,
(3) A measurable function
-integrable if
We note that Eq. (36) implies
where
for all
we easily get the following theorem.
Theorem 3..5. (1) A monotone interval-valued 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 interval-valued set function and
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 non-negative measurable function
Theorem 3..6. ([23, Lemma 2.5 (i) and (v)]) Let
a monotone interval-valued function. If
is continuous from above and we take
for all
(1)
(2)
where
Note that Theorem 3.6(2) implies the following equation (36) under the same condition of
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
Theorem 3..7. Let
a monotone interval-valued set function. If
is continuous and
Proof. Let
is continuous from above, by (40), we have
Since
is continuous from below and
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 interval-valued set function.
Theorem 3..8. Let
be a monotone interval-valued set function and let
is continuous from above, then we have
Proof. If
If
Theorem 3..9. Let a monotone interval-valued set function
be continuous from above and let
-integrable function. If
is continuous from above and
Proof. Since
is a monotone interval-valued set function, by Theorem 3.3 (1) and (2),
By Eq. (36) and Eq. (42), we have the result.
We remark that if we take a
-integrable function
is not monotone, that is, for each pair
Theorem 3..10. Let
be a monotone intervalvalued set function which is continuous from above, and let
-integrable functions with
Proof. The proof is similar to the proof of Theorem 3.10.
Theorem 3..11. Let
be monotone interval-valued set functions,
-integrable and
-integrable function, and
(1) If
then we have
(2) If
then we have
Proof. (1) Note that
if and only if
By (36) and (47), we have the result.
(2) Note that
if and only if
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 interval-valued set functions.
Theorem 3..12. Let (
with
are measurable functions, then we have
Proof. Since
is a submodular monotone interval-valued set function, by Theorem 3.3(6),
By Theorem 2.5, we have
and
By Eq. (36), eq. (50), and eq. (51), we have the result.
Theorem 3..13. Let (
with
are measurable functions with disjoint support, then
Proof. Since
is a subadditive monotone interval-valued set function, by Theorem 3.3(5),
and
By Eq. (36), Eq. (53), and Eq. (54), we have the result.
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 interval-valued set function of measurable functions.
From Theorem 3.2, Definition 3.3(3), and the condition that
of measurable functions
-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 bi-Choquet integral with respect to a monotone interval-valued set function
No potential conflict of interest relevant to this article was reported.