Image measures of infinite product measures and generalized Bernoulli convolutions

We examine measure preserving mappings $f$ acting from a probability space $(\mathrm{\Omega },F,\mu )$ into a probability space $$ where ${\mu }^{\ast }=\mu (f^{-1})$. Conditions on $f$, under which $f$ preserves the relations ‘‘to be singular’’ and ‘‘to be absolutely continuous’’ between measures defined on $(\mathrm{\Omega },F)$ and corresponding image measures, are investigated. We apply the results to investigate the distribution of the random variable $$ where $% \lambda \in (0;1),$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 $\psi = \sum\limits^{\infty}_{k=1} \xi _ka_k,$where$a_k>0$ are terms of the convergent series.