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Funktsional. Anal. i Prilozhen., 2002, Volume 36, Issue 2, Pages 12–27 (Mi faa187)  

This article is cited in 22 scientific papers (total in 23 papers)

Classification of Measurable Functions of Several Variables and Invariantly Distributed Random Matrices

A. M. Vershik

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: The classification of measurable functions of several variables is reduced to the problem of describing some special measures on the matrix (tensor) space, namely, the so-called matrix (tensor) distributions, that are invariant with respect to the permutations of indices. In the case of functions with additional symmetries (symmetric, unitarily or orthogonally invariant, etc.), these measures also have additional symmetries. This relationship between measurable functions and measures on the tensor space as well as our method in itself are used in both directions, namely, on one hand, to investigate invariance properties of functions and characterizations of matrix distributions, and, on the other hand, to classify the set of all invariant measures. We also give a canonical model of a measurable function with a given matrix distribution.

Keywords: classification of functions, matrix distributions, infinite symmetric group


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English version:
Functional Analysis and Its Applications, 2002, 36:2, 93–105

Bibliographic databases:

UDC: 513.5
Received: 26.12.2001

Citation: A. M. Vershik, “Classification of Measurable Functions of Several Variables and Invariantly Distributed Random Matrices”, Funktsional. Anal. i Prilozhen., 36:2 (2002), 12–27; Funct. Anal. Appl., 36:2 (2002), 93–105

Citation in format AMSBIB
\by A.~M.~Vershik
\paper Classification of Measurable Functions of Several Variables and Invariantly Distributed Random Matrices
\jour Funktsional. Anal. i Prilozhen.
\yr 2002
\vol 36
\issue 2
\pages 12--27
\jour Funct. Anal. Appl.
\yr 2002
\vol 36
\issue 2
\pages 93--105

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    This publication is cited in the following articles:
    1. Vershik A.M., “Random metric space is the universal Urysohn space”, Dokl. Math., 66:3 (2002), 421–424  mathnet  mathscinet  zmath  isi
    2. A. M. Vershik, “Kolmogorov's example (a survey of actions of infinite-dimensional groups with an invariant probability measure)”, Theory Probab. Appl., 48:2 (2004), 373–378  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. A. M. Vershik, “The Kantorovich metric: initial history and little-known applications”, J. Math. Sci. (N. Y.), 133:4 (2006), 1410–1417  mathnet  crossref  mathscinet  zmath  elib  elib
    4. J. Math. Sci. (N. Y.), 141:6 (2007), 1601–1607  mathnet  crossref  mathscinet  elib
    5. Vershik A.M., “Universality and randomness for the graphs and metric spaces”, Frontiers in Number Theory, Physics and Geometry I - ON RANDOM MATRICES, ZETA FUNCTIONS, AND DYNAMICAL SYSTEMS, 2006, 245–266  crossref  mathscinet  zmath  isi
    6. A. M. Vershik, “Does There Exist a Lebesgue Measure in the Infinite-Dimensional Space?”, Proc. Steklov Inst. Math., 259 (2007), 248–272  mathnet  crossref  mathscinet  zmath  elib  elib
    7. Vershik A.M., “Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of Urysohn space”, Topology Appl., 155:14 (2008), 1618–1626  crossref  mathscinet  zmath  isi  elib  scopus
    8. Petrov F., Vershik A., “Uncountable graphs and invariant measures on the set of universal countable graphs”, Random Structures & Algorithms, 37:3 (2010), 389–406  crossref  mathscinet  zmath  isi  scopus
    9. A. M. Vershik, “Scailing entropy and automorphisms with pure pointspectrum”, St. Petersburg Math. J., 23:1 (2012), 75–91  mathnet  crossref  mathscinet  zmath  isi  elib
    10. A. M. Vershik, “On classification of measurable functions of several variables”, J. Math. Sci. (N. Y.), 190:3 (2013), 427–437  mathnet  crossref  mathscinet
    11. Vershik A.M., Zatitskiy P.B., Petrov F.V., “Geometry and Dynamics of Admissible Metrics in Measure Spaces”, Cent. Eur. J. Math., 11:3 (2013), 379–400  crossref  mathscinet  zmath  isi  elib  scopus
    12. A. M. Vershik, P. B. Zatitskii, F. V. Petrov, “Virtual Continuity of Measurable Functions of Several Variables and Embedding Theorems”, Funct. Anal. Appl., 47:3 (2013), 165–173  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. 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
    14. J. Math. Sci. (N. Y.), 200:6 (2014), 677–681  mathnet  crossref
    15. 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
    16. A. M. Vershik, “Standardness as an Invariant Formulation of Independence”, Funct. Anal. Appl., 49:4 (2015), 253–263  mathnet  crossref  crossref  isi  elib
    17. J. Math. Sci. (N. Y.), 219:5 (2016), 683–699  mathnet  crossref  mathscinet
    18. Ackerman N., Freer C., Nesetril J., Patel R., “Invariant Measures Via Inverse Limits of Finite Structures”, Eur. J. Comb., 52:B (2016), 248–289  crossref  mathscinet  zmath  isi  elib  scopus
    19. Vershik A.M., “Asymptotic theory of path spaces of graded graphs and its applications”, Jap. J. Math., 11:2 (2016), 151–218  crossref  mathscinet  zmath  isi  scopus
    20. Ackerman N., Freer C., Patel R., “Invariant Measures Concentrated on Countable Structures”, Forum Math. Sigma, 4 (2016), e17  crossref  mathscinet  zmath  isi
    21. 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
    22. A. M. Vershik, “Asimptotika razbieniya kuba na simpleksy Veilya i kodirovanie skhemy Bernulli”, Funkts. analiz i ego pril., 53:2 (2019), 11–31  mathnet  crossref  elib
    23. L. N. Coregliano, A. A. Razborov, “Semantic limits of dense combinatorial objects”, Russian Math. Surveys, 75:4 (2020), 627–723  mathnet  crossref  crossref
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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