Classification of measurable functions of several variables and matrix distributions

Abstract

We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, it is an invariant of this function with respect to a certain group of transformations of variables; on the other hand, there is a special probability measure in the space of matrices (tensors) that is invariant under actions of natural infinite permutation groups. The intricate interplay of both interpretations of matrix (tensor) distributions makes them an important subject of modern functional analysis. We state and prove a theorem saying that, under certain conditions on a measurable function of two variables, its matrix distribution is a~complete invariant.