Classification of Measurable Functions of Several Variables and Invariantly Distributed Random Matrices

The classification of measurable functions of several variables is reduced to the problem of describing some special measures on the matrix (tensor) space, namely, the so-called matrix (tensor) distributions, that are invariant with respect to the permutations of indices. In the case of functions with additional symmetries (symmetric, unitarily or orthogonally invariant, etc.), these measures also have additional symmetries. This relationship between measurable functions and measures on the tensor space as well as our method in itself are used in both directions, namely, on one hand, to investigate invariance properties of functions and characterizations of matrix distributions, and, on the other hand, to classify the set of all invariant measures. We also give a canonical model of a measurable function with a given matrix distribution.