Vershik, A. M.; Zatitskiy, P. B.; Petrov, F. V. Virtual continuity of measurable functions of several variables and embedding theorems. (English. Russian original) Zbl 1305.28004 Funct. Anal. Appl. 47, No. 3, 165-173 (2013); translation from Funkts. Anal. Prilozh. 47, No. 3, 1-11 (2013). Summary: 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. Cited in 5 Documents MSC: 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 26B05 Continuity and differentiation questions 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:admissible metric; virtual continuity; function of several variables; polymorphism; trace theorem PDFBibTeX XMLCite \textit{A. M. Vershik} et al., Funct. Anal. Appl. 47, No. 3, 165--173 (2013; Zbl 1305.28004); translation from Funkts. Anal. Prilozh. 47, No. 3, 1--11 (2013) Full Text: DOI arXiv References: [1] L. V. Kantorovich, ”On the Translocation of Masses,” Dokl. Akad. Nauk SSSR, 37:7–8 (1942), 227–229; English transl.: J. Math. Sci., 133:4 (2006), 1381–1382. · Zbl 0061.09705 [2] V. G. 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