|
This article is cited in 9 scientific papers (total in 9 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.
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
Linking options:
https://www.mathnet.ru/eng/rm9628https://doi.org/10.1070/RM2014v069n06ABEH004927 https://www.mathnet.ru/eng/rm/v69/i6/p81
|
Statistics & downloads: |
Abstract page: | 982 | Russian version PDF: | 432 | English version PDF: | 41 | References: | 63 | First page: | 39 |
|