Set-valued Choquet integrals revisited

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

“…Definition 2.1 ( [2,4,5]). Let X be a nonempty set, Ω be a σ-algebra of subsets of X, and µ : Ω → [0, ∞) be a nonnegative real-valued set function.…”

“…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].…”

A Multi-Valued Choquet Integral with Respect to a Multisubmeasure

“…In [22], using the Kuratowski convergence, Zhang, Guo and Liu studied some properties of the set-valued Choquet integral of multifunctions with respect to real fuzzy measures and proved some convergence theorems for this kind of integral.…”

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

“…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.…”