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Zap. Nauchn. Sem. POMI, 2006, Volume 334, Pages 57–67 (Mi znsl222)  

This article is cited in 3 scientific papers (total in 4 papers)

Compactness of the congruence group of measurable functions in several variables

A. M. Vershika, U. Haböckb

a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
b University of Vienna

Abstract: We solve a problem, which appears in functional analysis and geometry, on the group of symmetries of functions of several arguments. Let $f\colon\prod_{i=1}^n X_i\longrightarrow Z$ be a measurable function defined on the product of finitely many standard probability spaces $(X_i,\frak B_i,\mu_i)$, $1\le i\le n$, that takes values in any standard Borel space $Z$. We consider the Borel group of all $n$-tuples $(g_1,…,g_n)$ of measure preserving automorphisms of the respective spaces $(X_i,\frak B_i,\mu_i)$ such that $f(g_1x_1,…,g_nx_n)=f(x_1,…,x_n)$ almost everywhere and prove that this group is compact, provided that its ‘trivial’ symmetries are factored out. As a consequence, we are able to characterise all groups that result in such a way. This problem appears with the question of classifying measurable functions in several variables, which has been solved in [2] but is interesting in itself.

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English version:
Journal of Mathematical Sciences (New York), 2007, 141:6, 1601–1607

Bibliographic databases:

UDC: 519.2
Received: 09.10.2006
Language: English

Citation: A. M. Vershik, U. Haböck, “Compactness of the congruence group of measurable functions in several variables”, Computational methods and algorithms. Part XIX, Zap. Nauchn. Sem. POMI, 334, POMI, St. Petersburg, 2006, 57–67; J. Math. Sci. (N. Y.), 141:6 (2007), 1601–1607

Citation in format AMSBIB
\Bibitem{VerHab06}
\by A.~M.~Vershik, U.~Hab\"ock
\paper Compactness of the congruence group of measurable functions in several variables
\inbook Computational methods and algorithms. Part~XIX
\serial Zap. Nauchn. Sem. POMI
\yr 2006
\vol 334
\pages 57--67
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl222}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2270907}
\elib{http://elibrary.ru/item.asp?id=9304139}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2007
\vol 141
\issue 6
\pages 1601--1607
\crossref{https://doi.org/10.1007/s10958-007-0068-7}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33846959194}


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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, “On classification of measurable functions of several variables”, J. Math. Sci. (N. Y.), 190:3 (2013), 427–437  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. J. Math. Sci. (N. Y.), 219:5 (2016), 683–699  mathnet  crossref  mathscinet
    4. Lovasz L., Szegedy B., “the Automorphism Group of a Graphon”, J. Algebra, 421:SI (2015), 136–166  crossref  mathscinet  zmath  isi  elib  scopus
  • Записки научных семинаров ПОМИ
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