关于${ \bf R}上的Moore-Yamasaki-Kharazishvili型测度和Borel扩散概率测度的无穷幂
On Moore-Yamasaki-Kharazishvili type measures and the infinite powers of Borel diffused probability measures on ${\bf R}
摘要：
本文简要介绍了山崎显着的调查（1980年）之间的关系摩尔-山崎-Kharazishvili型措施和无限权力的Borel扩散概率测度的${\bf R}$。更确切地说，我们给山崎的证明，没有无穷大权力的严格正密度函数的波莱尔概率测度$R$有一个等价的摩尔山崎Kharazishvili型措施。山崎的例子的某种修改是用于建设这样一个摩尔山崎Kharazishvili型措施，相当于产品的某个无限家庭的Borel概率测度与严格的正密度函数$R$。利用实轴上等分布序列的性质，证明了在$R$上任意一族具有严格正密度函数的无穷幂Borel扩散概率测度是强分离的，并相应地具有未知分布函数的无穷样本良基估计.这推广了[ Zerakidze Z.，Pantsulaia G.，萨塔什维利湾关于Borel和Baire族的分离问题$G$ -$\mathbb{R}$上的移位测度的幂//乌克兰数学。J。-2013。- 65（4）.-第470- 485页]。
英文摘要：
The paper contains a brief description of Yamasaki's remarkable investigation (1980) of the relationship between Moore-Yamasaki-Kharazishvili type measures and infinite powers of Borel diffused probability measures on ${\bf R}$. More precisely, we give Yamasaki's proof that no infinite power of the Borel probability measure with a strictly positive density function on $R$ has an equivalent Moore-Yamasaki-Kharazishvili type measure. A certain modification of Yamasaki's example is used for the construction of such a Moore-Yamasaki-Kharazishvili type measure that is equivalent to the product of a certain infinite family of Borel probability measures with a strictly positive density function on $R$. By virtue of the properties of equidistributed sequences on the real axis, it is demonstrated that an arbitrary family of infinite powers of Borel diffused probability measures with strictly positive density functions on $R$ is strongly separated and, accordingly, has an infinite-sample well-founded estimator of the unknown distribution function. This extends the main result established in [ Zerakidze Z., Pantsulaia G., Saatashvili G. On the separation problem for a family of Borel and Baire $G$-powers of shift-measures on $\mathbb{R}$ // Ukrainian Math. J. -2013.-65 (4).- P. 470--485 ].
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