We develop a new method for the construction of metric, probabilistic and dimensional theories for families of representations of real numbers via studies of special mappings, under which symbols of a given representation are mapped into the same symbols of other representation from the same family, and they preserve the Lebesgue measure and the Hausdorff-Besicovitch dimension (for such mappings the set of points of discontinuity can be everywhere dense). These mappings are said to be G-mappings (G-isomorphisms of representations). Probabilistic, metric and dimensional theories of G-isomorphic representations are identical. We show a rather deep connection between the faithfulness of systems of coverings, generated by different representations, and the preservation of the Hausdorff-Besicovitch dimension of sets by the above mentioned mappings.
