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As powerful tools for modeling nondeterministic problems, Choquet integrals have attracted much attention in recent years. This study generalizes the existing set-valued Choquet integrals and formulates the theory of Choquet integrals with respect to set-valued fuzzy measures. First, a set-valued Choquet integral of a real-valued function is defined using the set of Choquet integrals of real-valued functions with respect to fuzzy measure selections of a set-valued fuzzy measure. The integrand is then extended to set-valued functions, after which various properties, convergence theorems, and set-valued Jensen's inequalities are obtained. Second, following how single-valued Choquet integrals are defined, an alternative set-valued Choquet integral of real-valued function is formulated using Aumann integrals. Through these steps, the existing set-valued Choquet integrals are covered. [ABSTRACT FROM AUTHOR]

A constructively k -additive set function μ defined on a measurable space (X , B) is represented by using some σ -additive measure μ [ k ]. In this paper, we define the base measurable space for μ [ k ] on which μ [ k ] is uniquely defined. Using this measure, we define the covering measure μ ¯ to be useful in the analysis of the measurable space. We elucidate that a measurable set A is a strong μ -null set (μ (B) = μ (B ∪ C) , ∀ B , ∀ C ⊂ A) if and only if μ ¯ (A) = 0. We prove the subadditivity of μ ¯ and this follows the linearity and completeness of L p -spaces with respect to the Choquet and Pan integrals over μ ¯. The equivalent classes of these function spaces are given by the relation "to take same value on the complement of a strong μ -null set." [ABSTRACT FROM AUTHOR]

In this study, we generalize the concept of the k -order additivity of a set function. First, we discuss the Möbius transform for a non-discrete set function. Next, we generalize the definition of the k -order additivity of a set function using the Möbius transform and provide the equivalent conditions for the k -order additivity. Furthermore, we consider the k -order additivity of the distorted monotone measure. We prove that under certain conditions, a distorted measure is k -order additive if and only if the distortion function is a polynomial of k -th order. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we extend the concept of set-valued fuzzy measures (introduced and studied by Guo and Zhang [2004. On set-valued fuzzy measures. Information Sciences 160 (2004) 13–25]) to the class of multivalued set functions taking values in the family of non-empty closed subsets of a real normed space. Also, we introduce the notions of atom and non-atomicity of a multivalued set function and present some of their properties. We also establish an extension result by preserving non-atomicity for a fuzzy multisubmeasure/multimeasure. [Copyright &y& Elsevier]