Jensen's inequality for Choquet integral revisited and a note on Jensen's inequality for generalized Choquet integral

doi:10.1016/j.fss.2021.09.004

Fuzzy Sets and Systems

Volume 430, 28 February 2022, Pages 79-87

https://doi.org/10.1016/j.fss.2021.09.004 Get rights and content

Abstract

In this paper, it is shown that Jensen's inequality for Choquet integral given by R. Wang ten years ago is incorrect, then it is revisited, the modified Jensen's and reverse Jensen's inequalities for Choquet integral are proved. Then Jensen's inequality for generalized Choquet integral, obtained by the authors in a recent paper, is modified accordingly.

Section snippets

Preliminaries and introduction

The Choquet integral [1] was introduced as a useful tool for modeling non-deterministic problems. For an overview on the theory the reader is referred to Denneberg [2], Pap [6], [7] and Wang and Klir [9]. For the up-to-date information on the latest developments on the generalization of the Choquet integral, the reader is referred to Dimuro et al. [3]. We recall the basic concepts at first.

Let $R =] - \infty, \infty [$ , $R^{+} = [0, \infty [$ and $(X, A)$ be a measurable space. A set function $μ : A \to R^{+} \cup {\infty}$ is a fuzzy measure [9]

Jensen's inequality for Choquet integral revisited

Let $(X, A, μ)$ be a fuzzy measure space. First we give a counter-example to Theorem 1.3.

Counter-example 2.1. Let $X = [0, 1]$ , $A = B o r e l ([0, 1])$ , $μ = λ^{2}$ , where λ is the Lebesgue measure. Taking $f (x) = x$ , $φ (t) = {(t - 1)}^{2}$ , $x, t \in [0, 1]$ , then we see that φ is a non-increasing convex function. We calculate $(C) \int_{X} f d μ = (C) \int_{X} x d μ = \int_{0}^{1} λ^{2} ((x ⩾ α) \cap [0, 1]) d α = \int_{0}^{1} {(1 - α)}^{2} d α = \frac{1}{3}$ and $φ ((C) \int_{E} f d μ) = {(\frac{1}{3} - 1)}^{2} = \frac{4}{9}$ On the other side we have $(C) \int_{X} (φ \circ f) d μ = (C) \int_{X} {(x - 1)}^{2} d μ = \int_{0}^{1} λ^{2} (({(x - 1)}^{2} ⩾ α)) \cap [0, 1]) d α = \int_{0}^{1} {(1 - \sqrt{α})}^{2} d α = \frac{1}{6}$ Then we obtain $φ ((C) \int_{X} f d μ) = \frac{4}{9} > \frac{1}{6} = (C) \int_{X} (φ \circ f) d μ .$

A note on Jensen's inequality for generalized Choquet integral

In this section, we continue to use the notions and notations from [10]. The structure $([a, b], \oplus, \otimes)$ is a g-semiring with a generator $g : [a, b] \to [0, \infty]$ , i.e., $x \oplus y = g^{- 1} (g (x) + g (y)), x \otimes y = g^{- 1} (g (x) \cdot g (y)), \forall x, y \in [a, b]$ and $(X, A)$ is a measurable space, $μ : A \to [a, b]$ is a fuzzy measure.

The generalized Choquet integral of a measurable function $f : X \to [a, b]$ over $E \in A$ is given by $(C_{g}) \int_{E}^{\oplus} f \otimes d μ = (C) \int_{E} (g \circ f) d (g \circ μ) .$ Let $U \subseteq [a, b]$ be a $(\oplus, \otimes)$ -convex set.

Definition 3.1

([11]) Let $([a, b], \oplus, \otimes)$ be a semiring. A function $φ : U \to [a, b]$ is said to be $(\oplus, \otimes)$

Concluding remarks

The mistakes in Jensen's inequality for Choquet integral by Wang [8] are corrected. Then one of our previous result on Jensen's inequality for generalized Choquet integral is revised. It is necessary to point out that Jensen's inequality in higher dimension (Theorem 4.4 [8]) is also incorrect. There are some different discussions on this topic, the reader is referred to Mesiar et al. [5], Jang [4].

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 work was supported by the National Natural Science Fund of China (No. 11271062) and the Natural Science Fund of Jilin Province (No. 20190201014JC) (for the first author), by grants APVV-18-0052 (for the second author), and by Science Fund of the Republic of Serbia, Grant No. 652410, AI-ATLAS (for the third author).

References (12)

G.P. Dimuro et al.
The state-of-art of the generalizations of the Choquet integral: from aggregation and pre-aggregation to ordered directionally monotone functions
Inf. Fusion
(2020)
G. Choquet
Theory of capacities
Ann. Inst. Fourier
(1953)
D. Denneberg
Non-Additive Measure and Integral
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L.C. Jang
A note on Jensen type inequality for Choquet integrals
Int. J. Fuzzy Log. Intell. Syst.
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The Choquet integral as Lebesgue integral and related inequalities
Kybernetika
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E. Pap
Null-Additive Set Functions
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Short communicationJensen's inequality for Choquet integral revisited and a note on Jensen's inequality for generalized Choquet integral

Abstract

Section snippets

Preliminaries and introduction

Jensen's inequality for Choquet integral revisited

A note on Jensen's inequality for generalized Choquet integral

Concluding remarks

Declaration of Competing Interest

Acknowledgements

Inf. Fusion

Theory of capacities

Ann. Inst. Fourier

Non-Additive Measure and Integral

A note on Jensen type inequality for Choquet integrals

Int. J. Fuzzy Log. Intell. Syst.

The Choquet integral as Lebesgue integral and related inequalities

Kybernetika

Null-Additive Set Functions

Short communication
Jensen's inequality for Choquet integral revisited and a note on Jensen's inequality for generalized Choquet integral