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Uspekhi Mat. Nauk, 2014, Volume 69, Issue 6(420), Pages 81–114 (Mi umn9628)  

This article is cited in 4 scientific papers (total in 4 papers)

Virtual continuity of measurable functions and its applications

A. M. Vershikabc, P. B. Zatitskiybd, F. V. Petrovab

a St. Petersburg State University
b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)
d Chebyshev Laboratory at St. Petersburg State University

Abstract: A 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 a product of sets having almost full measure) does not hold in general. The search for a correct analogue of Luzin's theorem leads to a notion of virtually continuous functions of several variables. This apparently 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. The authors' results imply that under the conditions of Sobolev theorems there is a well-defined integration of a function with respect to a wide class of singular measures, including measures concentrated on submanifolds. The notion of virtual continuity is also used for the classification of measurable functions of several variables and in some questions on dynamical systems, the theory of polymorphisms, and bistochastic measures. In this paper the necessary definitions and properties of admissible metrics are recalled, several definitions of virtual continuity are given, and some applications are discussed.
Bibliography: 24 titles.

Keywords: admissible metrics, virtual topology, bistochastic measures, trace theorems, embedding theorems.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00373-а
13-01-12422-офи-м
Ministry of Education and Science of the Russian Federation MK-6133.2013.1
11.G34.31.0026
Gazprom Neft
Saint Petersburg State University 6.38.223.2014
This work was supported by the Russian Foundation for Basic Research (grant nos. 14-01-00373-a and 13-01-12422-офи-м), grant no. MK-6133.2013.1 of the President of the Russian Federation, the Chebyshev Laboratory at St. Petersburg State University (grant no. 11.G34.31.0026 of the Government of the Russian Federation), JSC Gazprom Neft, and St. Petersburg State University (grant no. 6.38.223.2014).


DOI: https://doi.org/10.4213/rm9628

Full text: PDF file (747 kB)
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English version:
Russian Mathematical Surveys, 2014, 69:6, 1031–1063

Bibliographic databases:

Document Type: Article
UDC: 517.37
MSC: Primary 28A20, 26B05; Secondary 54E35, 46E35
Received: 29.10.2014

Citation: A. M. Vershik, P. B. Zatitskiy, F. V. Petrov, “Virtual continuity of measurable functions and its applications”, Uspekhi Mat. Nauk, 69:6(420) (2014), 81–114; Russian Math. Surveys, 69:6 (2014), 1031–1063

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. St. Petersburg Math. J., 27:3 (2016), 393–398  mathnet  crossref  mathscinet  isi  elib
    2. A. M. Vershik, “Asymptotic theory of path spaces of graded graphs and its applications”, Jap. J. Math., 11:2 (2016), 151–218  crossref  mathscinet  zmath  isi  scopus
    3. A. M. Vershik, “The theory of filtrations of subalgebras, standardness, and independence”, Russian Math. Surveys, 72:2 (2017), 257–333  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. V. I. Bogachev, A. N. Kalinin, S. N. Popova, “O ravenstve znachenii v zadachakh Monzha i Kantorovicha”, Veroyatnost i statistika. 25, Posvyaschaetsya pamyati Vladimira Nikolaevicha SUDAKOVA, Zap. nauchn. sem. POMI, 457, POMI, SPb., 2017, 53–73  mathnet
  • Успехи математических наук Russian Mathematical Surveys
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