The Generalized Integrals Of Fuzzy Number Valued Functions

In this thesis the generalized integrals of fuzzy number valued functions are studied.The thesis consists of the following six chapters.In chapter 1,introduction to some basic problems are demonstrated,including the background,significance,the situation of current study related to fuzzy integral both domestic and abroad.In chapter 2,the related preliminary knowledge are presented.In chapter 3,it is proved that the integral of the decreasing function defined by Denneberg via distribution function is equivalent to the Lebesgue integral of the same function.In chapter 4,by using measurable function to define the symmetric Choquet integral of interval valued function,it is proved that the symmetric choquet integral of an interval valued function has basic properties such as monotonicity,triangular inequality etc.Then,the symmetric Choquet integral of fuzzy number valued function is defined.By example of classical integral,it is shown that the symmetric Choquet integral of a fuzzy number valued functions may not be a fuzzy number.The condition when the symmetric Choquet integral value of a fuzzy number valued function is a fuzzy number is studied.It follows that the result of the integral is a fuzzy number if the symmetric Choquet integral is finite.Some basic properties of the integral are further studied.In chapter 5,the convergence of symmetric Choquet integrals of fuzzy number valued functions under levelwise convergence are studied,and the convergence of symmetric Choquet integral of fuzzy number valued functions based on the supremum metric are also studied.Finally in chapter 5,the autocontinuity of the symmetric Choquet integral of fuzzy number valued function are discussed.It follows that the autocontinuity from above(below)of the integral is the same as the original fuzzy measure defining the integral.In chapter 6,the main contents of the thesis are summarized.Some topics related to the thesis worthy further studied are proposed.