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

TL;DR

This paper explores the relationship between Moore-Yamasaki-Kharazishvili type measures and infinite powers of Borel diffused probability measures on the real line, providing new constructions and demonstrating strong separation properties.

Contribution

It proves that no infinite power of a Borel probability measure with positive density is equivalent to a Moore-Yamasaki-Kharazishvili measure, and constructs an equivalent measure using a modified example.

Findings

No infinite power of a Borel measure with positive density is equivalent to a Moore-Yamasaki-Kharazishvili measure.

A modified example constructs an equivalent Moore-Yamasaki-Kharazishvili measure.

Families of such measures are strongly separated and admit infinite-sample estimators.

Abstract

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 ].