  We resume the results from \cite{Vershik FA} on the classification of
measurable functions in several variables, with some minor corrections of
purely technical nature, and give a partial solution to the characterization
problem of so--called matrix distributions, which are the metric invariants of
measurable functions introduced in \cite{Vershik FA}. The characterization of
these invariants of the ergodic measures on the space of matrices is closely
related to Aldous' and Hoover's representation of row-- and column--exchangable
distributions \cite{Aldous1981,Hoover1982}, but not in such an obvious way as
was initially expected in \cite{Vershik FA}.
