Image measures of infinite product measures and generalized Bernoulli convolutions

Abstract

We examine measure preserving mappings $f$ acting from a probability space $(\Omega ,F,\mu )$ into a probability space where ${\mu }^{\ast }=\mu (f^{-1})$. Conditions on $f$, under which $f$ preserves the relations ''to be singular'' and ''to be absolutely continuous'' between measures defined on $(\Omega ,F)$ and corresponding image measures, are investigated. We apply the results to investigate the distribution of the random variable where and ${\xi }_k$ are independent not necessarily identically distributed random variables taking the values $i$ with probabilities ,$i=0,1.$ We also studied in details the metric-topological and fractal properties of the distribution of a random variable $\psi =\sum _{k=1}^{\infty }{\xi }_k{a}_k,$ where $a_k>0$ are terms of the convergent series.

15 pages