On preservation of singularity, absolute continuity and discreteness under transformation of probability spaces

Abstract

The paper is devoted to the study of conditions for the preservation of mutual singularity resp. absolute continuity, and discreteness of probability measures under measurable mappings of probability spaces. Under very general assumptions we have found such conditions for the preservations. At the same time a series of important counterexamples are presented. The results obtained can simplified essentially the study the Lebesgue structure (i.e., finding necessary and sufficient conditions for the singular continuity, absolute continuity and discreteness of a wide spectra of probability measures with independent digits of symbolic expansions of real numbers and their multidimensional generalizations.