Vershik’s intermediate level standardness criterion and the scale of an automorphism

Vershik’s standardness criterion takes a particular form of combinatorial nature in the case of $r_n$-adic filtrations, which we call Vershik’s intermediate level criterion in this paper. This criterion has been intensively used in the ergodic theory literature, but it is not easily applicable by probabilists because it is stated in a language proper to the theory of measurable partitions and has not been translated in probabilistic terms. We aim to provide an easily applicable probabilistic statement of this criterion. Finally, Vershik’s intermediate level criterion is illustrated by revisiting Vershik’s definition of the scale of an invertible measure-preserving transformation.