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

We resume the results from [Ver02a] 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 [Ver02a]. The characterization of these invariants, considered as SN × SN–invariant, ergodic measures on the space of matrices is closely related to Aldous’ and Hoover’s representation of row– and column–exchangable distributions [Ald81, Ho82], but not in such an obvious way as was initially expected in [Ver02a].