On Properties of the Choquet Integral of Interval‐Valued Functions

Abstract

Based on the concept of an interval-valued function which is motivated by the goal to represent an uncertain function, we define the Choquet integral with respect to a fuzzy measure of interval-valued functions. We also discuss convergence in the(C)mean and convergence in a fuzzy measure of sequences of measurable interval-valued functions. In particular, we investigate the convergence theorem for the Choquet integral of measurable interval-valued functions.

Some Characterizations of the Choquet Integral with Respect to a Monotone Interval-Valued Set Function

“…In [21], we can find the theorem below. This gives a useful and interesting tool for the application of the Choquet integral of a non-negative measurable function f , with respect to a monotone interval-valued set functionμ.…”

“…In the past decade, it has been suggested to use intervals in order to represent uncertainty, for example, for economic uncertainty [12], for fuzzy random variables [13], in intervalprobability [14], for martingales of multi-valued functions [15], in the integrals of set-valued functions [16], in the Choquet integrals of interval-valued (or closed set-valued) functions [17][18][19][20][21][22], and for interval-valued capacity functions [23]. Couso-Montes-Gil [24] studied applications under the sufficient and necessary conditions on monotone set functions, i.e., the subadditivity of the Choquet integral with respect to monotone set functions.…”

Fuzzified Choquet Integral and its Applications in MADM: A Review and A New Method

“…Before expressing CI, it is necessary to remind the ideas of Lebegue integral and Sugeno integral. Then, Choquet integral and some of its applications in MCDM problems is considered [21,36,39,42]. Let ðX; K; lÞ be a measure space and l be r-finite.…”

On Jensen-Type and Hölder-Type Inequality for Interval-Valued Choquet Integrals

“…The Choquet integrals have been studied by many researchers (see [1][2][3][4][5][6]). Aumann [7], Jang and his colleagues [8][9][10][11][12][13], and Zhang et al [14] also have been studying the interval-valued Choquet integrals which are related with some properties and applications of them.…”

“…Aumann [7], Jang and his colleagues [8][9][10][11][12][13], and Zhang et al [14] also have been studying the interval-valued Choquet integrals which are related with some properties and applications of them. Various integral inequalities, such as Jensen's inequality, Hölder's inequality, Minkowski's inequality, and Chebyshev's inequality for some integrals were developed by the authors in [3,5,15,16].…”