关于集值测度变差的Radon－Nikodym问题

On the Radon-Nikodym Problem for the Variation of Multimeasures

本文引进了对偶空间Ｘ值可测函数ｆ：Ω→Ｘ关于Ｂａｎａｃｈ空间Ｘ上集值测度Ｍ的Ｄｉｎｃｕｌｅａｎｕ型积分（Ｄ）ｆΩｆｄＭ，给出了该积分的另一等价形式，证明了集值测度变差｜Ｍ｜关于Ｍ在上述积分意义下的Ｒａｄｏｎ－Ｎｉｋｏｄｙｍ导数的存在性定理。当Ｍ退化为向量测度时，本文的结果可直接推出［１］中定理１。

Abstract In this paper,the Dinculeanu type integral(D)∫ΩfdM of a measurable dual space X-valued function with respect to a multimeasure valued in a Banach space X is introduced,and equivalent form Of this integral is also given.A Radon-Nikodym theorem for the variation|M| with respect to M in the sense of above integral is proved.when M degenerates to a vector measure,theorem 1 of[1]can be deduced directly from our Radon-Nikodym theroem.