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

We resume the results from [12] on the classification of measurable functions in several variables, with some minor corrections of purely technical nature. We give a partial solution of he characterization problem of so-called matrix distributions, which are the metric invariants of measurable functions introduced in [12]. Matrix distibutions considered as $\S_\mathbb N\times\S_\mathbb N$-invariant, ergodic measures on the space of matrices – this fact connects our problem with Aldous' and Hoover's theorem [2,6].