Since Choquet put forward the theory of capacity and its integral,a lot of researches have been done on the Choquet integral of real-valued functions with respect to fuzzy measures.So far many studies have been devoted to a discrete case since it widely applied in decision problems for the Choquet integral of set-valued functions with respect to fuzzy measures.In this article,based on a Class of distorted fuzzy measures,the analytic properties of the Choquet integral of set-valued functions with respect to fuzzy measures are studied.Firstly,based on the concept of distorted fuzzy measures,we consider the in-tegral representation of Choquet integral of set-valued functions respect to fuzzy measures.the Choquet integral of set-valued function is transformed into the real-valued Choquet integral with respect to fuzzy measure,and the Radon-Nikodym property is given.Secondly,we discuss the properties of the primitive functions of Choquet integral for set-valued functions respect to fuzzy measures,such as zero additivity,pseudo-metric properties,S-properties,and so on.Finally,we improve the new definition of choquet integral of set-valued function respect to fuzzy measure,defined the upper and below Choquet integral and interval value Choquet integral of set-valued function,and the representation theorem of the interval value Choquet integral of set-valued function is given,at the same time,the primitive function of interval value Choquet integral of set-valued function respect to fuzzy measures is discussed. |