Set Valued Measure And Set Valued Bartle Integral

Let (X, ||·||) be a Banach space with the dual X^* ,and let(Ω,Σ) be a measurable space ,M : Σ→ 2^X\{(?)} be a set valued measure. In this thesis,the definition that a set of vector valued measure is element composite is first introduced ,by this definition ,we prove that ,if X is a reflexive Banach space ,then for each element composite bounded set K in bvca(Σ,X), by M(A) (?) (co)|-{m(A) : m ∈ K},(?)A ∈ Σ , a bounded countably additive set valued measure from Σ to P_bfc(X) is defined ,and a relationship between set valued measure and vector valued measure is obtained.Then ,by a equivalent statements about the strongly additive set valued measure,we introduce the definition of uniform strongly additive for a collection strongly additive set valued measure,and establish the Vitali-Hahn-Saks-Nikodym theorem about set valued measure. In the third section ,firstly, we give a new set valued Bartle integral definition ,then explore the integral properties for scalar function with respect to a bounded closed and convex set valued measure and establishes the limit theorem for set valued Bartle integral.