A Study On Monotone Set-Valued Measure Spaces

In this paper,we main study the properties of generalized on monotone set-valued measure spaces.The following is our main work which consists of two parts:In the first part,as a special kind of monotone set function taking value in the power set of an m-dimension space,the concept of monotone set-valued measure is introduced,and its different kinds of continuity are defined.On these bases,for a sequence of measurable functions defined on monotone set-valued measure spaces,some convergence concepts such as: almost everywhere convergence,convergence in measure,almost uniform convergence everywhere are given.The relationships among these convergences are discussed.Some important results such as Lebesgue Theorem and Egoroff Theorem in the classical measure theory are generalized to monotone set-valued measure theory.In the second part,the concept of atoms and pseudo atoms on monotone set-valued measure space are put forward and the properties of them are discussed.Moreover,the concept of(N)integrals on monotone set-valued measure spaces are defined,some basic properties and important theorems of(N)integral are given.