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Classical Luzin's theorem states that the measurable function of one variableis "almost" continuous. This is not so anymore for functions of severalvariables. The search of right analogue of the Luzin theorem leads to a notionof virtually continuous functions of several variables. This probably newnotion appears implicitly in the statements like embeddings theorems and tracestheorems for Sobolev spaces. In fact, it reveals their nature as theorems aboutvirtual continuity. This notion is especially useful for the study andclassification of measurable functions, aswell as in some questions ondynamical systems, polymorphisms and bistochastic measures. In this work werecall necessary definitions and properties of admissible metrics, definevirtual continuity, describe some of applications. Detailed analysis is to bepresented in another paper.

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