Elsevier

Information Sciences

Volume 177, Issue 15, 1 August 2007, Pages 3192-3201
Information Sciences

A Jensen type inequality for fuzzy integrals

https://doi.org/10.1016/j.ins.2007.02.006Get rights and content

Abstract

In this paper, we show a Jensen type inequality for the Sugeno integral. We also discuss some conditions assuring the satisfaction of opposite inequality (reverse Jensen inequality).

Introduction

The theory of fuzzy measures and fuzzy integrals was introduced by Sugeno [18] as a tool for modeling non-deterministic problems.

The properties and applications of the Sugeno integral have been studied by many authors, including Ralescu and Adams [8], Román-Flores et al. [12], [13] and Liu et al. [5]. For an overview on fuzzy measure and fuzzy integration theory, the reader is referred to Wang and Klir [20].

In connection with the topics that will be discussed in this article, there are several other theoretical and applied papers related to fuzzy measure theory on metric spaces, such as Li et al. [2], [3], Narukawa et al. [6] and Song [17].

For an overview of general theory of non-additive set functions and its applications, the book Null-additive set functions by Pap [7] is an excellent resource. For some applications of fuzzy measures, we refer the reader to [4], [9], [16], [19].

Not long ago, authors in [10] analyzed an interesting type of geometric inequalities for the Sugeno integral with some applications to convex geometry. More precisely, a Prékopa–Leindler type inequality for fuzzy integrals was proven, and subsequently used for the characterization of some convexity properties of fuzzy measures.

The purpose of this paper is to study a Jensen type inequality for the Sugeno integral. In this context, the research has two main aims. One is a Jensen type inequality for fuzzy integrals. The other one provides sufficient conditions assuring the reverse Jensen inequality.

The paper is organized as follows. Some necessary preliminaries are presented in Section 2. We address the essential problems in Sections 3 A Jensen type inequality for fuzzy integrals, 4 . Concluding remarks are in Section 5.

Section snippets

Fuzzy measures and Sugeno integral

In the sequel, we present some definitions and basic properties of the Sugeno integral that will be used in the next sections.

Definition 1

Let Σ be a σ-algebra of subsets of X and let μ:Σ[0,] be a nonnegative, extended real-valued set function. We say that μ is a fuzzy measure if:

  • (a)

    μ()=0;

  • (b)

    E,FΣ and EF imply μ(E)μ(F) (monotonicity);

  • (c)

    {Ep}Σ,E1E2, imply limpμ(Ep)=μ(p=1Ep) (lower continuity);

  • (d)

    {Ep}Σ,E1E2,μ(E1)<, imply limpμ(Ep)=μp=1Ep (upper continuity).

For an overview on fuzzy measures and

A Jensen type inequality for fuzzy integrals

The classical Jensen’s inequality (see [15]) is the following mathematical property of convex functions:ΦfdμΦ(f)dμ,where f is μ-measurable and Φ:[0,)[0,) is a convex function.

Remark 3

We know that inequality (4) is deeply connected with the vectorial nature of the concepts of the Lebesgue integral and the convexity of functions. However, the Sugeno integral is defined by using the reticular structure of R+ and, consequently, we might not expect the validity of (4), under the same conditions, in

H-continuity and reverse Jensen type inequality

The aim of this section is to show some connections between H-continuity of fuzzy measures and the Jensen’s inequality. More precisely, we aim to find conditions assuring the satisfaction of the opposite inequality in (5) (reverse Jensen’s inequality). As we will see, a possible approximation to solving this problem can be done via H-continuity of fuzzy measures and level-continuity of functions.The concept of H-continuity of fuzzy measures was exhaustively studied by the authors in [9], and one

Concluding remarks

The classical Jensen’s inequality is a useful result in several theoretical and applied fields. It provides a fundamental tool for understanding and predicting consequences of variance in any dynamical system (i.e., in terms of the classical expectation). However, the expectation associated to a non-deterministic phenomena is naturally the Sugeno integral (see [18], [20]).

In this paper we have presented a Jensen’s type inequality for the Sugeno integral which is obtained by replacing the

Acknowledgement

The author gratefully acknowledges financial support from Conicyt-Chile through Projects Fondecyt 1040303 and 1061244. Finally, we are grateful to the referees for their critical reading of the manuscript and many useful suggestions.

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