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Funktsional. Anal. i Prilozhen., 2013, Volume 47, Issue 3, Pages 1–11 (Mi faa3121)  

This article is cited in 5 scientific papers (total in 6 papers)

Virtual Continuity of Measurable Functions of Several Variables and Embedding Theorems

A. M. Vershikab, P. B. Zatitskiiab, F. V. Petrovab

a Saint Petersburg State University
b St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences

Abstract: 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.

Keywords: admissible metric, virtual continuity, function of several variables, polymorphism, trace theorem.

DOI: https://doi.org/10.4213/faa3121

Full text: PDF file (191 kB)
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English version:
Functional Analysis and Its Applications, 2013, 47:3, 165–173

Bibliographic databases:

UDC: 519.217
Received: 30.05.2013

Citation: A. M. Vershik, P. B. Zatitskii, F. V. Petrov, “Virtual Continuity of Measurable Functions of Several Variables and Embedding Theorems”, Funktsional. Anal. i Prilozhen., 47:3 (2013), 1–11; Funct. Anal. Appl., 47:3 (2013), 165–173

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. M. Vershik, “Two ways to define compatible metrics on the simplex of measures”, J. Math. Sci. (N. Y.), 196:2 (2014), 138–143  mathnet  crossref  mathscinet
    2. V. M. Buchstaber, M. I. Gordin, I. A. Ibragimov, V. A. Kaimanovich, A. A. Kirillov, A. A. Lodkin, S. P. Novikov, A. Yu. Okounkov, G. I. Olshanski, F. V. Petrov, Ya. G. Sinai, L. D. Faddeev, S. V. Fomin, N. V. Tsilevich, Yu. V. Yakubovich, “Anatolii Moiseevich Vershik (on his 80th birthday)”, Russian Math. Surveys, 69:1 (2014), 165–179  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. A. M. Vershik, P. B. Zatitskiy, F. V. Petrov, “Virtual continuity of measurable functions and its applications”, Russian Math. Surveys, 69:6 (2014), 1031–1063  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. St. Petersburg Math. J., 27:3 (2016), 393–398  mathnet  crossref  mathscinet  isi  elib
    5. D. Zaev, “On the Monge–Kantorovich Problem with Additional Linear Constraints”, Math. Notes, 98:5 (2015), 725–741  mathnet  crossref  crossref  mathscinet  isi  elib
    6. D. A. Zaev, “On ergodic decompositions related to the Kantorovich problem”, J. Math. Sci. (N. Y.), 216:1 (2016), 65–83  mathnet  crossref  mathscinet
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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