Researches On Set-valued Measures And Set-valued Fuzzy Measures

In this dissertation, the theory of set-valued measures and set-valued fuzzy measures are investigated on the base of the theory of set-valued functions.First, on the subsets class of m-dimension positive Euclid spaces, a kind of order structure is introduced. Furthermore, the concept of the convergence of the sequence of sets in that order is defined. Then, some fundamental natures of this order are given and the uniqueness of the limit of the sequence of sets which is convergent is given also. In addition, the definitions of pseudo-null-additivity, pseudo-null-subtractivity, pseudo-autocontinuity and uniform pseudo-autocontinuity of set-valued fuzzy measures are defined and the containing relations of them are studied.In the second part, by the Hausdorff metric and the inclusion relationship of the sets, the set-valued fuzzy measure is defined on the subsets class of the real normed space of all non-empty bounded subsets. Sequentially, the concepts of the null-additivity, autocontinuity, uniformly autocontinuity, pseudo-null-additivity, pseudo-autocontinuity, uniformly pseudo-autocontinuity, and converse-autocontinuity are defined and the containing relations of them are investigated.In the third part, in the case that the Banach space is not reflexive, the Lebesgue decomposition for the set-valued measure with the value of bounded closed convex set is established. The results are the generalizations of the corresponding conclusions of the existing literatures.