On the classification problem of matrix distributions of measurable functions in several variables

TL;DR

This paper reviews and refines previous results on classifying measurable functions of multiple variables and offers a partial characterization of matrix distributions, linking them to exchangeable measures.

Contribution

It provides a partial solution to characterizing matrix distributions as invariants of measurable functions, building on and correcting earlier work.

Findings

Refined classification results for measurable functions in several variables.

Partial characterization of matrix distributions as metric invariants.

Connections established between matrix distributions and exchangeable measures.

Abstract

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}.