没有原子的概率测度空间的一个性质

A property of probability measure spaces with no atoms

设(X,μ)是一个没有原子的概率测度空间,则测度μ可视为由单位质量经反复细分所获得的测度.证明从(X,μ)到([0,1),m)的保测映射的存在性.作为这个结果的应用,给出了空间L2(X,μ)上的标准正交系的构造方法.最后,具体给出L2(C,μc)上的一个标准正交系,其中C是三分Cantor集,cμ是Cantor测度.

Let （X,μ） be a probability measure space with no atoms. Then μ can be regarged as a measure defined by repeated subdivision. It proved that there is a measure-preserving transformation from（X,/1） to （[0, 1）,m）, As an application of this result, gives a method for constructing an orthonormal system on L2 （X,μ）. Finally, obtains an orthonormal system on L2 （C,μc）, where C is the middle-third Cantor set, μc is the Cantor meausre.