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The paper contains a brief description of Yamasaki's remarkable investigation(1980) of the relationship between Moore-Yamasaki-Kharazishvili type measuresand infinite powers of Borel diffused probability measures on ${\bf R}$. Moreprecisely, we give Yamasaki's proof that no infinite power of the Borelprobability measure with a strictly positive density function on $R$ has anequivalent Moore-Yamasaki-Kharazishvili type measure. A certain modification ofYamasaki's example is used for the construction of such aMoore-Yamasaki-Kharazishvili type measure that is equivalent to the product ofa certain infinite family of Borel probability measures with a strictlypositive density function on $R$. By virtue of the properties ofequidistributed sequences on the real axis, it is demonstrated that anarbitrary family of infinite powers of Borel diffused probability measures withstrictly positive density functions on $R$ is strongly separated and,accordingly, has an infinite-sample well-founded estimator of the unknowndistribution function. This extends the main result established in [ ZerakidzeZ., Pantsulaia G., Saatashvili G. On the separation problem for a family ofBorel and Baire $G$-powers of shift-measures on $\mathbb{R}$ // Ukrainian Math.J. -2013.-65 (4).- P. 470--485 ].

On Moore-Yamasaki-Kharazishvili type measures and the infinite powers of Borel diffused probability measures on ${\bf R}