4. On Moore–Yamasaki–Kharazishvili Type Measures and the Infinite Powers of Borel Diffused Probability Measures on R

This chapter 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 \(\mathbf{R}\). More precisely, there is given 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 the properties of real-valued sequences equidistributed 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 the paper [ZPS].