Distorted Lebesgue Measures Research Articles

Computation of Choquet integrals: Analytical approach for continuous functions

In the continuous case, analytical computations of the Choquet integral are limited, despite being commonly used in various applications. One can either use the definition, which is computationally demanding and impractical, or apply already existing formulas restricted only to monotone nonnegative functions on a real interval starting at zero. This article aims to present more convenient computational formulas for continuous functions without imposing restrictions on their monotonicity given any real interval. First, a more general approach to monotone functions is provided for both positive and negative functions. Then, reordering techniques are introduced to compute the Choquet integral of an arbitrary continuous function, and with these, a monotone equivalent to every function can be constructed. This equivalent function preserves the final Choquet integral value, implying that only formulas for monotone functions are required. In addition to general fuzzy measures, the article assumes particular cases of distorted Lebesgue measures and distorted probabilities as the most commonly used fuzzy measures.

Existence and uniqueness of solutions of differential equations with respect to non‐additive measures

By taking Sugeno‐derivative into account, first, we investigate the existence of solutions to the initial value problems (IVP) of first‐order differential equations with respect to non‐additive measure (more precisely, distorted Lebesgue measure). It particularly occurs in the mathematical modeling of biology. We begin by expressing the differential equation in terms of ordinary derivative and the derivative with respect to the distorted Lebesgue measure. Then, by using the fixed point theorem on cones, we construct an operator and prove the existence of positive non‐decreasing solutions on cones in semi‐order Banach spaces. In addition, we also use Picard–Lindelöf theorem to prove the existence and uniqueness of the solution of the equation.Second, we investigate the existence of a solution to the boundary value problem (BVP) on cones with integral boundary conditions of a mix‐order differential equation with respect to non‐additive measures. Moreover, the Krasnoselskii fixed point theorem is also applied to both BVP and IVP and obtains at least one positive non‐decreasing solution. Examples with graphs are provided to validate the results.

Refinements of Jensen's inequalities for Choquet integrals and applications

In this paper, we investigate the refinements of Jensen's inequalities in Choquet calculus and applications. We propose respectively one refinement of Theorem 3.3 – Jensen type inequality I and four refinements of Theorem 3.4 – Jensen type inequality II in Wang's article (Wang, 2011 [10]), and then use these refinements to prove other inequalities. It is specially mentioned that Chebyshev's inequality, Hölder type inequality and Minkowski type inequality for Choquet integrals are proved more easily by using the refinements of Jensen type inequality in 2 dimensions. What's more, we provide some examples in the case of the distorted Lebesgue measure to illustrate our results.

Choquet integral analytic inequalities

Based on an amazing result of Sugeno [15], we are able to transfer classic analytic integral inequalities to Choquet integral setting. We give Choquet integral inequalities of the following types: fractional-Polya, Ostrowski, fractional Ostrowski, Hermite-Hadamard, Simpson and Iyengar. We provide several examples for the involved distorted Lebesgue measure.

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

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.

Choquet integrals of r-convex functions

This study explores the upper bound and the lower bound of Choquet integrals for r-convex functions. Firstly, we show that the Hadamard inequality of this kind of integrals does not hold, but in the framework of distorted Lebesgue measure we can provide the similar Hadamard inequality of Choquet integral as the r-convex function is monotone. Secondly, the upper bound of Choquet integral for general r-convex function is estimated, respectively, in the case of distorted Lebesgue measure and in the non-additive measure. Finally, we present two Jensen’s inequalities of Choquet integrals for r-convex functions, which can be used to estimate the lower bound of this kind, when the non-additive measure is submodular. What’s more, we provide some examples in the case of the distorted Lebesgue measure to illustrate all the results.

On a Choquet-Stieltjes type integral on intervals

Abstract In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

Fredholm-Choquet integral equations

In this paper we study the classical Fredholm integral equation of second kind, in which the Lebesgue type integral is replaced by the more general Choquet integral with respect to a monotone, submodular and continuous from below set function, including the so-called distorted Lebesgue measures.

Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials with Respect to Distorted Lebesgue Measures

For the univariate Bernstein-Kantorovich-Choquet polynomials written in terms of the Choquet integral with respect to a distorted probability Lebesgue measure, we obtain quantitative approximation estimates for the $L^{p}$-norm, $1\le p<+\infty$, in terms of a $K$-functional.

The Choquet integral of log-convex functions

In this paper we investigate the upper bound and the lower bound of the Choquet integral for log-convex functions. Firstly, for a monotone log-convex function, we state the similar Hadamard inequality of the Choquet integral in the framework of distorted measure. Secondly, we estimate the upper bound of the Choquet integral for a general log-convex function, respectively, in the case of distorted Lebesgue measure and in the non-additive measure. Finally, we present Jensen’s inequality of the Choquet integral for log-convex functions, which can be used to estimate the lower bound of this kind when the non-additive measure is concave. We provide some examples in the framework of the distorted Lebesgue measure to illustrate all the results.

A Way to Choquet Calculus

In this paper, we deal with the Choquet integral and derivative with respect to fuzzy measures on the nonnegative real line and present a way to Choquet calculus as a new research paradigm. In Choquet calculus, a representation for calculating the continuous Choquet integral is first given by restricting the integrand to a class of nondecreasing and continuous functions and the fuzzy measure to a class of distorted Lebesgue measures. Next, the derivative of functions with respect to distorted Lebesgue measures is defined as the inverse operation of the Choquet integral. Then, elementary properties in Choquet calculus are explored. In addition, we clarify the relation of Choquet calculus with fractional calculus, where the fractional Choquet integral and derivative are newly defined. In addition, we consider differential equations with respect to distorted Lebesgue measures and give their solutions. Finally, we introduce conditional distorted Lebesgue measures and explore their properties.

On the f-divergence for non-additive measures

The f-divergence evaluates the dissimilarity between two probability distributions defined in terms of the Radon–Nikodym derivative of these two probabilities. The f-divergence generalizes the Hellinger distance and the Kullback–Leibler divergence among other divergence functions. In this paper we define an analogous function for non-additive measures. We discuss them for distorted Lebesgue measures and give examples. Examples focus on the Hellinger distance.

ON PROPERTIES OF DERIVATIVES WITH RESPECT TO FUZZY MEASURES

Fuzzy measures in this paper are a class of distorted Lebesgue measures (see [5]). We investigate some properties of the derivatives with respect to distorted Lebesgue measures and give some important examples.

Choquet Integral of Fuzzy-Number-Valued Functions: The Differentiability of the Primitive with respect to Fuzzy Measures and Choquet Integral Equations

This paper deals with the Choquet integral of fuzzy-number-valued functions based on the nonnegative real line. We firstly give the definitions and the characterizations of the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on the nonadditive measure. Furthermore, the operational schemes of above several classes of integrals on a discrete set are investigated which enable us to calculate Choquet integrals in some applications. Secondly, we give a representation of the Choquet integral of a nonnegative, continuous, and increasing fuzzy-number-valued function with respect to a fuzzy measure. In addition, in order to solve Choquet integral equations of fuzzy-number-valued functions, a concept of the Laplace transformation for the fuzzy-number-valued functions in the sense of Choquet integral is introduced. For distorted Lebesgue measures, it is shown that Choquet integral equations of fuzzy-number-valued functions can be solved by the Laplace transformation. Finally, an example is given to illustrate the main results at the end of the paper.

A note on derivatives of functions with respect to fuzzy measures

This paper deals with the Choquet integral on the non-negative real line. First it gives a representation of the Choquet integral of a non-negative, continuous and increasing function with respect to a fuzzy measure. Next, restricting fuzzy measures to a class of distorted Lebesgue measures, it considers Choquet integral equations. In order to solve Choquet integral equations, a concept of the derivatives of functions with respect to fuzzy measures is introduced. For distorted Lebesgue measures, it is shown that Choquet integral equations are formulated as Volterra integral equations of the first kind. The differentiability of functions with respect to fuzzy measures is also discussed. It further shows a relation of a Choquet integral equation with the Abel integral equation. Finally this paper introduces simple differential equations with respect to fuzzy measures and gives their solutions.