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

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.