Elsevier

Information Sciences

Volume 630, June 2023, Pages 252-270
Information Sciences

Choquet type integrals for single-valued functions with respect to set-functions and set-multifunctions

https://doi.org/10.1016/j.ins.2023.02.038Get rights and content

Abstract

Due to their numerous applications such as in decision making, information fusion, game theory, and data mining, Choquet integrals have recently attracted much attention. In this study, two generalization types of Choquet integrals are presented. First, a generalized Choquet type integral of a single-valued function is introduced with respect to a set-function and measure. Several of its properties, such as convergence theorems and Jensen's inequality, are proved. Second, in the spirit of the single-valued Choquet integral, a generalized Choquet type set-valued integral for a single-valued function with respect to a set-multifunction and measure is introduced using Aumann integrals as well as various properties, including convergence theorems.

Introduction

As the core of classical analysis, integral theory has laid the foundation for other mathematical branches with many applications in theory and practice. There are two possibilities: single-valued integrals (i.e., for integrating scalar functions with respect to vector measure or vector functions with respect to scalar measure, and for considering the bilinear integral such as Lebesgue integrals [17], Bartle integral [3]) and set-valued integrals (i.e., the integrals take values in the power set of space X such as with the Aumann integral of multifunctions (set-valued functions) with respect to (w.r.t.) nonnegative real-valued σ-additive measures [1] and Bartle integral of single-valued functions w.r.t. multimeasures [29], [30], [40]). The set-valued integral has become very important in fields like mathematical economy and optimization theory. In order to manage fuzzy and nondeterministic problems, various kinds of nonadditive integrals, such as the Sugeno integral [35], Choquet integral [5], Shilkret integral [33], and pseudo-integral [26] were introduced in recent years. For an overview of the theory regarding nonadditive integrals, readers may refer to Pap [26], Wang and Klir [39], Torra et al. [37], and Zhang [47]. This study focuses on two generalization types of Choquet integrals: the Choquet type integral for a single-valued function w.r.t. a set-function and measure, and the Choquet type set-valued integral for a single-valued function w.r.t. a set-multifunction and measure. The Choquet integral [5], which was defined by French mathematician, Gustave Choquet, in 1953, has been frequently applied in decision making, information fusion, game theory, and data mining (see Candeloro [4], Sambucini [31]), thus it has attracted much attention in recent years. For more details, readers may refer to Denneberg [6], Grabisch [15], and Pap [26], as well as Dimuro et al. [7] for recent advancements.

Consider a measurable space (X,Σ) and a fuzzy measure (F-measure) ν:Σ[0,] (meets ν()=0 and monotonicity ν(U)ν(V) when UV). Suppose that h:X[0,] is a measurable function, then the classical Choquet integral is offered by:(Ch)hdν=0ν(h>β)dλ, in which λ refers to the standard Lebesgue measure. Generalizations of Choquet integrals are separated into two classes: single-valued and set-valued cases.

1. Single-valued case Some Choquet integral generalizations of this type are as follows:

(i) The Choquet integral w.r.t. a non-monotonic F-measure (see Murofushi et al. [24]) which replaces ν by a non-monotonic F-measure in Eq. (1.1).

(ii) The Choquet-Stieltjes integral (see Gal [8], Narukawa [25], Sun [36]) replaces λ by a Lebesgue-Stieltjes measure in Eq. (1.1).

(iii) The generalized Choquet integral (see Zhang et al. [46], Pap [28], Mesiar [22]) replaces ([0,],+,) by a semiring ([a,b],,) in Eq. (1.1).

2. Set-valued case Some Choquet integral generalizations of this type are as follows:

(i) The set-valued Choquet integral I (see Jang [18], Zhang et al. [44], Wang [38]) replaces the integrand h by a multifunction in Eq. (1.1).

(ii) The set-valued Choquet integral II (see Pap [27], Sofian-Boca [34], Gong [14]) replaces ν by an interval-valued submeasure in Eq. (1.1).

(iii) The set-valued Choquet integral III (see Zhang and Guo [42]) replaces h and ν, respectively, by a multifunction as well as a set-valued F-measure in Eq. (1.1).

These set-valued Choquet integrals are all defined using selections of either Aumann [1] or Papageorgiou [29], [30], which are the set of Choquet integrals for selectors of multifunctions or set-valued F-measures, respectively.

In the previous study directly related to this work [46], the GCI (generalized Choquet integral) based on semirings ([a,b],,) is as follows:(C)hd(ν,m)=[a,b]+ν(hβ)dm, where h is a nonnegative measurable function, ν is a F-measure, m is a σ-⊕-additive measure, and the right-hand integral is a pseudo-integral [26].

It had been shown that the GCI can cover many kinds of nonadditive integrals such as Sugeno integral, Choquet integral, Shilkret integral, and Choquet-Stieltjes integral. The GCI is for a monotonic F-measure which does not cover the non-monotonic F-measure Choquet integral defined by Murofushi and Sugeno [24]. Since the non-monotonic F-measure and its Choquet integral are very important and valuable in real-world applications [24], the subject requires further elaboration.

This study can be regarded as a continuation of the former work [46] which shall consider the semiring ([0,],+,) as well as replace F-measure ν in Eq. (1.2) by a set-function ϕ and a set-multifunction Φ. Then, the generalized Choquet type integral (GCTI) and generalized Choquet type set-valued integral (GCTSI) are obtained.

This present study includes two sections, as shown in Section 2, which will further abandon the condition “ν()=0” in non-monotonic F-measure since it can be viewed as an initial condition that is allowed to take a non-null value to then replace F-measure ν by an arbitrary set-function ϕ in Eq. (1.2). The resulting GCI in “(C)hd(ϕ,m),” covers the Choquet integral for non-monotonic F-measure, namely, GCTI. Some properties of the integral, such as convergence theorems and Jensen's inequalities, are proved. In section 3, ϕ will be replaced in the GCTI by a set-multifunction Φ. Then, a more general integral, namely, GCTSI in “(C)hd(Φ,m),” is determined (i.e., (C)hd(Φ,m)=[0,]Φ(h>β)dm), in which the right-hand side integral is in Aumann's sense.

Throughout this paper, the following symbols will be used:

R=],[;

R¯=[,];

R+=[0,[;

R¯+=[0,];

P0(R)={AR:A};

P¯0(R+)= P0(R+){};

Pf(k)(c)(R)={AP0(R):Aisclosed(compact)(convex)};

Pf(k)(c)(R+) and P¯f(k)(c)(R+) are obvious.

Section snippets

Choquet type integrals of single-valued functions with respect to set-functions

In this section, previous results in Choquet integrals and Choquet-Stieltjes integrals are extended, while the GCTI for functions w.r.t. set-functions and measures is introduced wherein some of its properties and different types of convergence theorems are proved. Then, a new Jensen's inequality is demonstrated after showing that its standard version does not hold.

Let (X,Σ) be a measurable space in which X represents a nonempty set and Σ refers to σ-algebra subsets of X. Also, (R+,B(R+)) is the

Choquet type integrals of single-valued functions with respect to set-multifunctions

In this section, the GCTSI of single-valued functions w.r.t. set-multifunctions is defined. This is an extension of Sofian-Boca's integral [34] since it was defined only for the interval-valued submeasure. Its various properties and convergence theorems will be proven.

Concluding remarks

(1) The GCTI of a single-valued function w.r.t. a set-function has been established. Some properties, convergence theorems, and Jensen's inequalities have also been provided. Therefore, this extends the existing Choquet integral in two aspects, the F-measure ν is extended to a set-function ϕ and the standard Lebesgue measure λ is extended to a measure m.

(2) The GCTSI of a single-valued function w.r.t. a set-multifunction has been defined based on GCTIs. The applied method is direct from Aumann

CRediT authorship contribution statement

Deli Zhang: Writing – original draft. Radko Mesiar: Writing – review & editing. Endre Pap: Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The present study was funded by the National Natural Science Foundation of China (No. 11271062) and the Natural Science Foundation of Jilin Province (No. 222614JC0106101856) (for the first author), by grant APVV-18-0052 and by the IGA project of the Faculty of Science Palacký University Olomouc IGAPRF2022017 (for the second author) as well as by the project on Artificial Intelligence ATLAS (grant No. 6524105) funded by Science Fund of the Republic Serbia (for the third author).

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