The Representation Of Choquet Integral Of The Set-valued Functions With Respect To Fuzzy Measures And The Characteristic Of Its Primitive

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.