Virtual Continuity of Measurable Functions and Its Applications

TL;DR

The paper introduces the concept of virtual continuity for measurable functions of several variables, extending classical results and enabling new applications in Sobolev space theory, measure integration, and dynamical systems.

Contribution

It proposes a new notion of virtual continuity for multivariable functions, connecting it to embedding theorems, measure integration, and dynamical systems applications.

Findings

Virtual continuity generalizes Luzin's theorem to multiple variables.

Allows integration over singular measures on submanifolds.

Links to Sobolev embedding and trace theorems.

Abstract

Classical theorem of Luzin states that a measurable function of one real variable is "almost" continuous. For measurable functions of several variables the analogous statement (continuity on the product of sets having almost full measure) does not hold in general. Searching for a right analogue of Luzin theorem leads to a notion of virtually continuous functions of several variables. This probably new notion implicitly appears in the statements of embedding theorems and trace theorems for Sobolev spaces. In fact it reveals the nature of such theorems as statements about virtual continuity. Our results imply that under conditions of Sobolev theorems there is a well-defined integration of a function over wide class of singular measures, including the measures concentrated on submanifolds. The notion of virtual continuity is used also for the classification of measurable functions of several variables and in some questions on dynamical systems, theory of polymorphisms and bistochastic measures. In this paper we recall necessary definitions and properties of admissible metrics, give several definitions of virtual continuity and discuss some applications. Revised version (without the proofs) is published in \cite{VZPFA}.