Virtual continuity of the measurable functions of several variables, and Sobolev embedding theorems

TL;DR

This paper introduces the concept of virtual continuity for measurable functions of several variables, connecting it to Sobolev embedding theorems and providing a new framework for analyzing such functions in various mathematical contexts.

Contribution

It proposes a novel notion of virtual continuity for multivariable functions, linking classical theorems to this new concept and exploring its applications.

Findings

Virtual continuity generalizes classical notions of almost continuity.

Connections established between virtual continuity and Sobolev embedding theorems.

Potential applications in dynamical systems and measure theory.

Abstract

Classical Luzin's theorem states that the measurable function of one variable is "almost" continuous. This is not so anymore for functions of several variables. The search of right analogue of the Luzin theorem leads to a notion of virtually continuous functions of several variables. This probably new notion appears implicitly in the statements like embeddings theorems and traces theorems for Sobolev spaces. In fact, it reveals their nature as theorems about virtual continuity. This notion is especially useful for the study and classification of measurable functions, aswell as in some questions on dynamical systems, polymorphisms and bistochastic measures. In this work we recall necessary definitions and properties of admissible metrics, define virtual continuity, describe some of applications. Detailed analysis is to be presented in another paper.