We study a problem of equivalent definitions of the Hausdorff-Besicovitch dimension via a system Ф(Q[infinity]) of cylinders of the Q[infinity]-expansion. Sufficient conditions for the system Ф(Q[infinity]) to be faithful for the determination of the Hausdorff-Besicovitch dimension on the unit interval are found and fine fractal properties of probability measures with independent (generally speaking non-identically distributed) Q[infinity]-digits are studied.