On classification of measurable functions of several variables

TL;DR

This paper introduces a canonical form and a complete invariant system for measurable functions of multiple variables, linking these concepts to matrix distributions and advancing the classification of such functions.

Contribution

It defines a new normal form and a complete invariant system for measurable functions, connecting these to matrix distributions for better classification.

Findings

Introduces the canonical image for measurable functions.

Describes a new complete system of invariants based on joint distributions.

Relates invariants to the matrix distribution, a previously known invariant.

Abstract

We define a normal form (called the canonical image) of an arbitrary measurable function of several variables with respect to a natural group of transformations; describe a new complete system of invariants of such a function (the system of joint distributions); and relate these notions to the matrix distribution, another invariant of measurable functions found earlier, which is a random matrix. Bibliography: 7 titles.