We show that the idea of spectral regularization introduced by Talagrand in the study of covering numbers of averages of contractions in a Hilbert space H can be concentrated in one inequality which turns out to be a suitable tool for the study of other characteristics of the set of averages. This inequality generates an intrinsic Lipschitz embedding of the circle and yields many useful corollaries. We also easily deduce the original Talagrand estimate of covering numbers and provide better estimates for geometric subsequences of the averages. Using majorizing measuring technique, we prove a new criterion of the a.s. convergence of random sequences under suitable incremental conditions. We obtain as a corollary the classical theorem of Rademacher-Menshov on orthogonal series and the famous spectral criterion for the strong law of large numbers due to Gaposhkin.