On the equality of integrals

doi:10.1016/j.ins.2017.02.015

Information Sciences

Volume 393, July 2017, Pages 82-90

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Abstract

Considering a finite space X, several necessary conditions and one rather general sufficient condition describing when the Choquet integral coincides with the pan-integral with respect to the standard arithmetic operations are shown. These conditions are characterized by using the minimal atoms of monotone measure. Under the introduced constraints, the calculation of these coinciding two integrals is also given.

Introduction

The Choquet integral [4], the pan-integral [32] and the concave integral [14] are three kinds of prominent nonlinear integrals with respect to monotone measure (or capacity), see, for example [3]. All these integrals have numerous application in economy, social sciences, data fusion, multicriteria decision support, etc., see, for example, [8], [10], [18]. It is well known that for the σ-additive measures all the three types of integrals coincide with the Lebesgue integral (i.e., these three integrals can be seen as particular generalizations of the Lebesgue integral). All these integrals can be seen as particular instances of decomposition integrals [6] (see also [23], [24], [25]). However, in general case they are significantly different from each other [22], [24], [25]. Recall that the concave integral is the greatest decomposition integral, while the pan-integral and the Choquet integral are incomparable, in general [13].

In [14] the relationship between the concave integral and the Choquet integral was discussed, and the concave integral was shown to coincide with the Choquet integral if and only if the underlying monotone measure m is convex (also known as supermodular) (see also [1], [16]).

Recently we discussed the relationship between the concave integral and the pan-integral on finite spaces [26]. We introduced the concept of minimal atom of a monotone measure. By using the characteristic of minimal atoms we presented a necessary and sufficient condition that the concave integral coincides with the pan-integral with respect to the usual arithmetic operations $+$ and· on finite spaces.

This paper will focus on the relationship between the Choquet integrals and pan-integrals on finite spaces. By means of minimal atoms of a monotone measure we show several necessary conditions and a sufficient condition that the Choquet integral coincides with the pan-integral w.r.t. the usual addition $+$ and usual multiplication ·. This characterizes monotone measures for which the related Choquet integrals and pan-integrals coincide. Under the introduced constraints, the calculation of these two coinciding integrals is also given.

Observe that the equality of general pan-integrals and Choquet-like integrals [19] is shortly discussed in Conclusions.

Section snippets

Preliminaries

Let X be a nonempty set and $A$ a σ-algebra of subsets of X, and $(X, A)$ denote a measurable space. A set function $m : A \to [0, + \infty [$ is called a monotone measure [2], [13], [30], if it satisfies the conditions: (1) $m (\emptyset) = 0$ and m(X) > 0; (2) m(A) ≤ m(B) whenever A ⊂ B and $A, B \in A$ .

A monotone measure m is said to be superadditive if $m (A \cup B) \geq m (A) + m (B)$ for any $A, B \in A$ and $A \cap B = \emptyset$ [5]; supermodular if $m (A \cup B) + m (A \cap B) \geq m (A) + m (B)$ for any $A, B \in A$ [5].

The concept of a pan-integral was introduced in [32] and it involves two

Coincidences of the Choquet and pan-integrals on finite spaces

In the rest of the paper, consider with no loss of generality, $X = {1, 2, \dots, n}$ as a fixed finite space for some integer $n \in N,$ and let $M_{n}$ be the class of all monotone measures on X, m: 2^X → [0, ∞[.

For the convenience of our discussion, we denote ${Pan}_{m} (f) = \int^{p a n} f d m,$ ${Ch}_{m} (f) = (C) \int f d m$ and ${Cav}_{m} (f) = \int^{c a v} f d m$ .

Our goal is to investigate monotone measures $m \in M_{n}$ such that the related pan and Choquet integrals coincide, i.e., ${Pan}_{m} (f) = {Ch}_{m} (f)$ for each f: X → [0, ∞[. Obviously, this happens whenever m is additive, i.e.,

The equality of the Choquet, pan and concave integrals

Recall that Lehrer in [14] has characterized all monotone measures $m \in M_{n}$ for which the Choquet and concave integral coincide.

Proposition 4.1

[14] Let $m \in M_{n}$ . Then Cav_m ≡ Ch_m if and only if m is supermodular, i.e., for any A, B ⊂ X it holds $m (A \cup B) + m (A \cap B) \geq m (A) + m (B) .$

Recently, we have characterized in [26] the conditions on $m \in M_{n}$ when the concave and pan-integrals coincide.

Proposition 4.2

Let $m \in M_{n}$ . Then Cav_m ≡ Pan_m if and only if the following two conditions holds:

(i)
m possesses the m-minimal atoms disjointness property, i.e., any pair

Conclusions

We have shown several necessary conditions and a sufficient condition for which the Choquet integral coincides with the pan-integral on finite spaces. Such conditions were characterized by minimal atoms of monotone measure (Theorems 3.4 and 3.10). Observe that in multicriteria decision support, as well as in the game theory, the disjointness of considered groups of criteria (of players) is rather often considered, which when evaluating optimal expected value based on a monotone measure yields

Acknowledgments

This work was supported by the grant APVV-14-0013, the National Natural Science Foundation of China (Grants No. 11371332 and No. 11571106) and the NSF of Zhejiang Province (No. LY15A010013).

The authors are grateful to the anonymous reviewers and editors for valuable comments helping us to improve the original version of this paper.

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On the equality of integrals

Abstract

Introduction

Section snippets

Preliminaries

Coincidences of the Choquet and pan-integrals on finite spaces

The equality of the Choquet, pan and concave integrals

Conclusions

Acknowledgments

Fuzzy Sets Syst.

J. Math. Anal. Appl.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inf. Sci.

J. Math. Anal. Appl.

Fuzzy Sets Syst.

Int. J. Approx. Reasoning

Int. J. Approx. Reasoning

J. Math. Anal. Appl.

J. Math. Anal. Appl.

Fuzzy Sets Syst.

Extendable cooperative games

J. Public Econ. Theory

Monotone set functions-based integrals

General theory of the fuzzy integral

Mathware Soft Comput.

Theory of capacities

Ann. Inst. Fourier