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 X be a non-empty 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 semi-continuous) 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 semi-continuous) 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 null-additive if

Definition 2..2. [2,3,5,24] Let μ be a monotone set function on A. (1) If f : X → ？⁺ is a non-negative measurable function, then the Choquet integral of f with respect to μ is defined by

where

for all α ∈ 2 ？⁺ and the integral on the right-hand side is the Lebesgue integral of μ_f .

(2) If f : X → ？ is a real-valued 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 ∈ ？⁺ and

for all α ∈ ？^？ = (？∞, 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 ？, and f a measurable function from X to ？.

(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 ？.

Definition 2..4. [24] Let μ be a monotone set function on A and I ⊂ ？ an index set. A class of real-valued 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 → ？⁺ is submodular and f, g : X → ？ are realvalued measurable functions, then we have

Theorem 2..6. Let (X, A) be a measurable space. If a monotone set function μ : A → ？⁺ is subadditive and f, g : X → ？ are measurable functions with disjoint support, that is, {x ∈ X | f(x) > 0}∩{x ∈ X | g(x) > 0} = Ø, then we have

In this section, we consider intervals, interval-valued functions, and the Aumann integral of measurable interval-valued functions. Let I(？) be the class of all bounded and closed intervals (intervals, for short) in ？ as follows:

For any a ∈ ？, we define a = [a, a]. Obviously, a ∈ I(？) [18-21].

Recall that if (？, ？, m) is the Lebesgue measure space and C(？) is the set of all closed subsets of ？, then the Aumann integral of a closed set-valued function F : ？ → C(？) 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) ？ Fdm is a nonempty bounded and closed interval in ？ whenever F is an interval-valued function as in the following theorem.

Theorem 3..1. If an interval-valued function F = [g_l, g_r] : ？ → I(？) is measurable and integrably bounded, then g_l, g_r ∈ S(F) and

where the two integrals on the right-hand side are the Lebesgue integral with respect to m.

Note that we write

for all bounded continuous function g. Let C(？) be the class of all closed ？. We recall that the Hausdorff metric d_H : C(？) × C(？) → ？⁺ is defined by

for all A, B ∈ C(？). It is well-known that for all ？ = [a_l,a_r],

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 {A_n}⊂ A and A ∈A such that A_n ↑ A, then

that is,

(4) An interval-valued 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 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 A, B ∈ A, then

(7) An interval-valued set function

is said to be submodular if A, B ∈ A, then

(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 μ_l and μ_r are set functions, and μ_l ≤ μ_r.

(2) An interval-valued set function

is monotone if only only if the set functions μ_l and μ_r are monotone.

(3) An interval-valued 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 interval-valued 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 interval-valued 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 interval-valued 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 interval-valued set function

is null-additive if and only if the set functions μ_l and μ_r are null-additive, and μ_l ≤ μ_r.

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 f : X → ？⁺, with respect to a monotone interval-valued set function

is defined by

where m is the Lebesgue measure on ？ and the integral on the right-hand side is the Aumann integral with respect to m of

(2) The Choquet integral of a real-valued measurable function f : X → ？, with respect to a monotone interval-valued 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 ∈ ？⁺. By the definition of

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 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 non-negative measurable function f, with respect to a monotone interval-valued set function

Theorem 3..6. ([23, Lemma 2.5 (i) and (v)]) Let f be a nonnegative measurable function and

a monotone interval-valued function. If

is continuous from above and we take

for all α ∈ ？⁺, 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 → ？, with respect to a continuous monotone interval-valued set function.

Theorem 3..7. Let f be a measurable function and

a monotone interval-valued 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 interval-valued set function.

Theorem 3..8. Let

be a monotone interval-valued 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 interval-valued set function

be continuous from above and let f a non-negative

-integrable function. If

is continuous from above and A, B ∈ A with A ⊂ B, then we have

Proof. Since

is a monotone interval-valued set function, by Theorem 3.3 (1) and (2), μ_l and μ_r are monotone interval-valued 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 non-negative

-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 interval-valued set functions, f a non-negative

-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 interval-valued set functions.

Theorem 3..12. Let (X, A) be a measurable space. If a continuous monotone interval-valued set function

with μ_l(X) = μ_r(X), is submodular and

are measurable functions, then we have

Proof. Since

is a submodular monotone interval-valued 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 interval-valued 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 interval-valued 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.

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 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 interval-valued set function

of measurable functions f : X → ？. 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 interval-valued set function on the space of non-negative

-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.