中文摘要：

给定有限测度空间（Ω，A，μ），令Mx（A）=span{^n∑＋i=1=χ_Ai x_i,A_i,x_i∈X,n∈N} L∞（μ,X）．证明了（Ω,A）上的向量值有限可加测度m是可列可加的当且仅当其对应泛函Ｕ是W＊-序列连续的，对应关系由U（x）／∫_Ω^xdm（x∈M_x（A））确定．并借助于向量值测度的Yosida—Hewitt分解定理，进一步证明了任一定义于M_x（A）上的连续线性泛函均能唯一分解成w＊-序列连续泛函与纯连续泛函的ι1-和．

英文摘要：

For a finite measure space （（Ω,A,μ）,let Mx（A） be the space of the uniform limits of the form ∑χ_Ai x_i （finite sum） withA, A_i∈A and xi∈ X. In this paper we show that a sufficient and necessary condition for a finitely additive vector-valued measure m on（Ω,A）to be countably additive is that the corresponding functional U defined by U（x） = ∫Ωx Jdm （for x∈Mx（A）） is w＊ -sequentiallycontinuous. With help of the Yosida-Hewitt decomposition theorem of vector-valued case,we show consequently that every continu-ous linear functional on Mx （A） can be uniquely decomposed into the 11-sum of a w＊-sequentially continuous functional and a purely continuous functional.