向量值函数McShane积分的收敛定理

The Convergence Theorems for McShane Integral of Vector valued Functions

实值函数的ＭｃＳｈａｎｅ积分是一种Ｒｉｅｍａｎｎ型绝对积分，它等价于Ｌｅｂｅｓｇｕｅ积分．向量值函数的ＭｃＳｈａｎｅ积分是实值函数ＭｃＳｈａｎｅ积分在Ｂａｎａｃｈ空间中的推广，它与实函数ＭｃＳｈａｎｅ积分有较大的差别．讨论了向量值函数ＭｃＳｈａｎｅ积分的收敛性问题，证明了一致收敛定理、平均收敛定理．特别地，当Ｘ＊的单位球＊弱列紧时，控制收敛定理也成立．

The McShane integral of real functions is a kind of Riemann type absolute integral which is equivalent to Lebesgue's one. It's generalization to Banach space, the McShane integral of vector valued functions is very different from itself. Here, the convergence problems of McShane integral of vector valued functions are discussed. The uniformly convergence theorem and mean convergence theorem are proved. Especially, the dominated convergence theorem is proved under the condition that the unit ball of X  is * weak sequencelly compact.