  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 ].
