Interval-valued Choquet integral for set-valued mappings: definitions, integral representations and primitive characteristics

Abstract: In this paper, a new kind of real-valued major Choquet integral, real-valued minor Choquet integral and interval-valued Choquet integrals for set-valued functions is introduced and investigated. The representations of the Choquet integral of set-valued functions with respect to a fuzzy measure are given. In particular, we focus on the case of the distorted Lebesgue measure as a fuzzy measure. Furthermore, the characteristics of the primitive of Choquet integral for set-valued functions are given as Radon-Nikodym property in some sense.


Introduction
The Choquet integral with respect to a fuzzy measure was proposed by Murofushi and Sugeno [13]. It was introduced by Choquet [4] in potential theory with the concept of capacity in 1950s. So far many studies have been devoted to a discrete case because their wide application in decision-making. In fact, there are many applications of the Choquet integral as an aggregation function, it has been used for utility theory in the field of economic theory [3,6], fuzzy reasoning [21], pattern recognition, information fusion and data mining [6,14], in particular, for multi-criteria decision making [7], and also as a psychological model of subjective preference [22], and so on. As Sugeno's description in [19], due to a wide range of applications for Choquet integral in decision problems, so far many studies have been devoted to a discrete case. However, the analytic properties of the Choquet integrals of set-valued functions with respect to fuzzy measure have not been fully discussed, including the properties of the primitives functions of Choquet integrals, the representation of Choquet integrals and the derivative of integral primitive functions in some sense, and so on. Indeed, integrals of set-valued function had been studied by Aumann [1]. Jang et al. [10] defined Choquet integrals of set-valued functions by using the measurable selection functions, that is by the Aumann's approach, and it has been discussed and generalized by Gong, Jang, Pap et al. [8,10,11,16,18]. In 2018, Paternain et al. [17] proposed a new approach for the interval-valued Choquet integral that takes into account every possible permutation fitting to the considered ordinal structure of data. Wu et al. [9] defined real-valued Choquet integrals for set-valued functions by where F is a measurable set-valued function, F α = {x|F(x) ∩ [α, ∞] ∅}, α ≥ 0. Specially, it could be computed by the Choquet integral of a real-valued function, that is ( It is well known that set-valued functions F(t) = [0, e t ] and G(t) = [t, e t ] defined on a compact convex set [0, 1] are quite different, but the integral values are equal by Eq (1) since their sup F(x) and sup G(x) are equal completely. Based on this consideration, we propose a new kind of real-valued major Choquet integral, real-valued minor Choquet integral and interval-valued Choquet integrals for setvalued functions, and investigate some properties, such as the representations of the Choquet integral of set-valued functions with respect to a fuzzy measure, the characteristics of the primitive of Choquet integral for set-valued functions as a Radon-Nikodym property in some sense, and so on. It shows that these results we obtained in this paper are consistent with the results of [10,11] for a compactconvex-set-valued function, and we notice that the method proposed in this paper is more simply than the calculations in [10,11].
This paper is organized as follows. Section 2 presents some concepts on fuzzy measures and Choquet integral of set-valued functions. Also, the main definition of Choquet integral of set-valued functions in this paper is given. Section 3 defines major and minor Choquet integrals for set-valued function and shows the basic transformation theorem. Section 4 shows the representation theorem of Choquet integrals for set-valued functions as a Radon-Nikodym properties. Section 5 discusses some properties of the primitives functions of Choquet integrals. Section 6 introduces a simple representation of Choquet integral for set-valued functions with respect to a distorted probability measure. Section 7 concludes this paper.

Preliminaries and Definitions
This section introduces some notations and basic concepts on fuzzy measures, Choquet integrals, set-valued functions and distorted fuzzy measure. Also, the main definition of Choquet integral of set-valued functions in this paper is given.
Let X be a nonempty set and A a σ−algebra on X. A fuzzy measure on X is a set function µ : A −→ [0, ∞] satisfying the following conditions [23]: (4) In X, if A 1 ⊃ A 2 · · · ⊃ A n ⊃ · · · , A n ∈ A, and there exist a n 0 , such that µ(A n 0 ) < ∞, then If µ is a fuzzy measure, then (X, A, µ) is said to be a fuzzy measure space. R + = [0, ∞] denotes the set of extended nonnegative real numbers. P(R + )\{∅} denotes the class of all the nonempty subsets of R + . Remark 2.1. Let m : R + −→ R + be a continuous and increasing function and m(0) = 0. A fuzzy measure µ = µ m , a distorted Lebesgue measure, is defined by µ m (·) = m(λ(·)). Similarly, let m : Definition 2.1. [4] Let (X, A, µ) be a fuzzy measure space, f : X −→ R + be a measurable real-valued function. Then the Choquet integral of f on A is defined as where f α = {x| f (x) ≥ α}, α ≥ 0, and the right part is the Lebesgue integral.
Remark 2.2. (i) Instead of (C) X Fdµ, we will write (C) Fdµ; (ii) If F is a Choquet integrably bounded set-valued function, then F is Choquet integrable; (iii) A set-valued function F : X −→ R + is said to be measurable if , then the above mentioned definition of the Choquet integral of set-valued functions is consistent with Definition 2.1. Furthermore, unless otherwise stated, in this paper, (C) , (c) , respectively denote the Choquet integral of set-valued functions, real-valued functions and the Lebesgue integral. Definition 2.3. Let (X, A, µ) be a fuzzy measure space, F : X −→ P(R + )\{∅} be a measurable setvalued function. Then major Choquet integral, minor Choquet integral, Choquet integrals of F on A are defined respectively by Thus, Definition 3.2 is consistent with the classical Choquet integral of real-valued functions.
It follows by Definition 2.1, Thus, by the Theorem 2.1, It follows by Definition 2.1, Thus, by the Theorem 2.1,  [11] for interval-valued functions, but we notice that the method proposed in this paper is simpler than the calculations in [11].

The representation and properties for the Choquet integral of the set-valued functions
Let µ be a distorted fuzzy measure and consider µ([τ, t]) for a closed interval[τ, t], then µ([τ, t]) is increasing for t and decreasing for τ. Throughout the paper, let m(t) and f (t) be continuously differentiable functions. Let µ([τ, t]) be differentiable with respect to τ on [0, t] for every t > 0. We require the regularity condition that µ({t}) = 0 holds for every respectively denotes major function and minor function of F(x). Furthermore, the definitions of major Choquet integral, minor Choquet integral and interval-valued Choquet integral for set-valued function, play important roles in discussing the problems concerning the representation theorem.
Theorem 3.1. Let F : X −→ P(R + )\{∅} be a measurable and strictly increasing set-valued function, . Then the Choquet integral of F(t) with respect to µ on [0, t] is represented as In particular, for µ = µ m , then Proof. For f + (x) = sup F(x). By Theorem 2.1, we have On the other hand, Similarly, we can also obtain . .
Hence, f − and f + is a constant function, Theorem 3.1 still holds. Therefore, we have the following corollary.

Radon-Nikodym properties of the Choquet integral for the set-valued functions
As Sugeno's description in article [19], due to a wide range of applications for Choquet integral in decision problems, so far many studies have been devoted to a discrete case. However, the analytic properties of the Choquet integrals of set-valued functions with respect to fuzzy measure have not been fully discussed, including the properties of the primitives functions of Choquet integrals, the representation of Choquet integrals and the derivative of integral primitive functions in some sense, and so on. It will be discussed in this section by using the definitions of real-valued major Choquet integral, real-valued minor Choquet integral and interval-valued Choquet integrals for set-valued functions.
Given two continuous increasing and nonnegative real-valued functions g 1 (x), g 2 (x) with g 1 (0) = g 2 (0) = 0, and a fuzzy measure µ m . How to define a continuous increasing and differentiable setvalued function F(x), such that Proof. By Theorem 3.1, we have Thus, we obtain On the other hand, The proof is complete. By the same way, we have Theorem 4.2. Let (X, A, µ) be a fuzzy measure space, g 1 (t) and g 2 (t) be nonnegative continuous realvalued functions. Then there exists F : [0, +∞) −→ P(R + )\{∅} being a measurable set-valued function, such that We just need to prove that f + is nonnegative continuous and differentiable increasing, and satisfying . In fact, since g 1 (t) is nonnegative continuous and differentiable on [0, +∞), so f + (t) is also nonnegative continuous and differentiable. It is easy to calculate that and Therefore, Similarly, make f − (t) = L −1 [G 1 (s)/sM(s)], we can also obtain

The primitive function characterization of the Choquet integral for set-valued functions
Definition 5.1. [12,15] Let (X, A, µ) be a fuzzy measure space.
(1) µ is said to be weakly null-additive if for any A, B ∈ A, µ(A) = µ(B) = 0 implies µ(A ∪ B) = 0; (2) µ is said to have strong order continuity if for any A n ⊂ A, A ∈ A with µ(A) = 0, A n ↓ A implies µ(A n ) −→ 0; (3) µ is said to have pseudo-metric generating property if for any ε > 0, there is δ > 0, such that µ(A ∪ B) < ε, whenever A, B ∈ A, and µ(A) ∨ µ(B) < δ; (4) µ is said to have property (S) if for any A n ⊂ A with µ(A n ) −→ 0, there exists a subsequence A n i of A n such that µ(lim i→∞ (A n i )) = 0. (2) ν is said to have strong order continuity if for any A n ⊂ A, (3) ν is said to have pseudo-metric generating property if for any In the following, let F be a nonnegative measurable set-valued function. For any A ∈ A, the set- in the case of the Lebesgue measure, µ(F α ∩ A) = 0 and µ(F α ∩ B) = 0 almost everywhere for α ∈ [0, +∞). By the monotonicity of the Lebesgue measure, we have ( By the dominated convergence theorem of the Lebesgue integral, we obtain Similarly, we can also obtain ν − ({A n }) = (C) A n Fdµ −→ 0, therefore, ν({A n }) −→ 0.
(3) ν is a fuzzy measure, we just need to prove that ν satisfies the following conditions: Similarly, we can also obtain Hence, ν(A) ≤ ν(B).
On the other hand, By the Lebesgue integral dominated convergence theorem, we have Similarly, we can also obtain ν On the other hand, By the Lebesgue integral dominated convergence theorem, we have Similarly, we can also obtain The proof is complete. Thus, in the case of the Lebesgue measures, µ(F α ∩ A) = 0 almost everywhere for α ∈ [0, +∞). By µ is strong order continuity, and µ(A) = 0, hence, µ(F α ∩ A n ) ↓ 0. On the other hand, since F(x) is Choquet integrable, we have By the Lebesgue integral dominated convergence theorem, we have By the same way, we can also obtain Fdµ] → 0, so ν has strong order continuity.
The proof is complete.
Theorem 5.4. Let (X, A, µ) be a fuzzy measure space, F be a measurable set-valued function and Choquet integrable on X, If µ has pseudo-metric generating property, then ν also has pseudo-metric generating property.
Hence, for any α ≥ 0, by the Lebesgue integral theory, measurable function sequence {µ(F α ∩ A n )} n −→ 0. According to Riesz Theorem from real analysis, there exists subsequence {µ(F α ∩ A n } k , such that By the monotony of µ, we obtain {µ(F α ∩ A n } k −→ 0. We suppose that µ(F α ∩ A n ) −→ 0(n → 0), then By the property (S ) of µ, so there exists a subsequence A (1) Repeating this procedure, we can obtain a subsequence A ( j) n i of An such that Let A n i = A ( j) n i , for any α > 0, there exists a with 1 2 a < α, and for α > 0, it follows that Similarly, we can also obtain ν − (lim i→∞ (A n i )) = 0. Therefore, ν(lim i→∞ (A n i )) = 0, so µ also has property (S ). The proof is complete.

The Choquet integral of the set-valued functions under distorted fuzzy measures
The properties of the Choquet integrals with respect to fuzzy measures have been studied extensively. This section we discussed the representation of the Choquet integral for set-valued functions with respect to a distorted fuzzy measure by probability measure p [2].  Since f + (x) is a continuous differentiable and not have plateaus, according to Lemma 6.1, there exists a monotone increasing function g + : X −→ X such that Similarly, we also obtain exists a monotone increasing function g − : X −→ X, such that On the other hand, due to F(x) = [ f − (x), f + (x)], we have Hence, there exists a monotone increasing set-valued function G(x) = [g − (x), g + (x)] with g + (x) = sup G(x) and g − (x) = inf G(x) , such that The proof is complete.

Conclusion
The aim of this paper is attempt to discuss the representation of Choquet integral of set-valued functions with respect to fuzzy measures and the characteristic of its primitive. We firstly define and discuss real-valued major Choquet integral, real-valued minor Choquet integral and intervalvalued Choquet integrals for set-valued functions with respect to fuzzy measures, which achieved the domination of the Choquet integral of set-valued functions with respect to fuzzy measures. Meanwhile, the calculation of the Choquet integral for set-valued function is investigated, and we have shown a basic representation theorem of Choquet integral for set-valued function as a Radon-Nikodym property in some sense. On the other hand, the characteristics of the primitive of the Choquet integral for set-valued functions are investigated. At the same time, we discussed the representation for Choquet integral of set-valued functions with respect to a distorted fuzzy measure by probability measure which is a classically fuzzy measure. Our results improve and generalize the corresponding results of [9]. In the future research, we shall discuss some applications for Choquet integral of set-valued functions with respect to fuzzy measures.