G. Pantsulaia | Searcharxiv

arXiv2016

On uniform distribution for invariant extensions of the linear Lebesgue measure

The concept of uniform distribution in $[0,1]$ is extended for a certain strictly separated maximal (in the sense of cardinality) family $(λ_t)_{t \in [0,1]}$ of invariant extensions of the linear Lebesgue measure $λ$ in $[0.1]$, and it is shown that the $λ_t^{\infty}$ measure of the set of all $λ_t$-uniformly distributed sequences is equal to $1$, where $λ_t^{\infty}$ denotes the infinite power of the measure $λ_t$. This is an analogue of Hlawka's (1956) theorem for $λ_t$-uniformly distributed sequences. An analogy of Weyl's (1916) theorem is obtained in similar manner.

math.CA↗

arXiv2015

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

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 ].

math.PR↗