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In a recent paper by the authors, Jensen's inequality for Choquet integral was given, and a wrong assertion—"Jensen's inequality does not hold for asymmetric Choquet integral" was made. This paper can be viewed as a continuation of the previous one, Jensen's inequality for asymmetric Choquet integral is proved, the error is corrected. As its generalization, Jensen's inequality for generalized asymmetric Choquet integral is obtained. [ABSTRACT FROM AUTHOR]

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]

Due to many applications, the Choquet integral as a powerful tool for modeling non-deterministic problems needs to be further extended. Therefore the paper is devoted to a generalization of the Choquet integral. As a basis, the pseudo-integral for bounded integrand is extended to the case for nonnegative integrands at first, and then the generalized Choquet integral is defined. As special cases, pseudo-Choquet Stieltjes integrals, pseudo-fuzzy Stieltjes integrals, g -Choquet integrals, pseudo-(N)fuzzy integrals and pseudo-(S)fuzzy integrals are obtained, and various kinds of properties and convergence theorems are shown, meanwhile Markov, Jensen, Minkowski and Hölder inequalities are proved. In the end, the generalized discrete Choquet integral is discussed. The obtained results for the generalized Choquet integral cover some previous results on different types of nonadditive integrals. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that Jensen's inequality for Choquet integral given by R. Wang ten years ago is incorrect, then it is revisited, the modified Jensen's and reverse Jensen's inequalities for Choquet integral are proved. Then Jensen's inequality for generalized Choquet integral, obtained by the authors in a recent paper, is modified accordingly. [ABSTRACT FROM AUTHOR]

This paper performs a further investigation on the set-valued Choquet integral given by Jang and Kwon (Fuzzy Sets and Systems 112 (2000) 233), some incorrect properties are revised. It is pointed out that the definition for convergence of sequence of sets or set-valued functions is unreasonable. By using the usual Kuratowski''s convergence, some new convergence theorems for the set-valued Choquet integral are shown. Meanwhile some other properties are obtained. [Copyright &y& Elsevier]