Virtual continuity of measurable functions of several variables and embedding theorems

Luzin’s classical theorem states that any measurable function of one variable is “almost” continuous. This is no longer true for measurable functions of several variables. The search for a correct analogue of Luzin’s theorem leads to the notion of virtually continuous functions of several variables. This, probably new, notion appears implicitly in statements such as embedding theorems and trace theorems for Sobolev spaces. In fact, it reveals their nature of being theorems about virtual continuity. This notion is especially useful for the study and classification of measurable functions, as well as in some questions on dynamical systems, polymorphisms, and bistochastic measures. In this work we recall the necessary definitions and properties of admissible metrics, define virtual continuity, and describe some of its applications. A detailed analysis will be presented elsewhere.