Radon-Nikodym Theorem Of Set Valued Measures In A Class Of Frechet Spaces

Set valued analysis is a branch of modern mathematics which has been developed rapidly and is an important component part of nonlinear analysis. Set valued measure is a set valued function that satisfies countable additivity and its value takes the subset of topological space as value. It is born and developed in the influence of the integral of set valued function. Set valued analysis has many applications in a lot of fields.The purpose of thesis is to discuss the relationship between the set valued measures which value is closed, convex and bounded set in Frechet space and two kind of integrals of set valued functions, that is, the Radon-Nikodym theorem of the set valued measures which value is closed, convex and bounded set. The relations between the distance deduced by the Hausdorff uniform topology and the Hausdorff distance deduced by the p in Frechet space is to be discussed from the analysis of the convergence of set-valued sequence. The Radon-Nikodym theorem of the set valued measures which value is closed, convex and bounded set is discussed when the measurable set valued function and integral are defined in the chapter 2 by the properties of set space without the embedding theorem[4]. The method of our proof is a generalization method used in proof of the theorems of the vector measure in Banach space in [26] and the correspondence theorem of the point-value measure of Frechet space in [6].The main results is as follows: let (Ω,Σ,μ)be a finitely complete metric space. X is a Frechet space satisfying preserve-direction conditions , M :Σ→P_bfc(X) is a set valued measure, then there exists an integrable set valued function F^～such that M(A)=∫_A F^～(ω)dμfor any A in I that is equivalent1) M isμ-continuous;2) M is a bounded variation;3) M has a locally compact average range, that is, given a A∈Σ⁺ andε>0 , there exists a B∈A∩Σ⁺ such thatμ(A\B)<εand is relatively compact.