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On the Classification Problem of Measurable Functions in Several Variables and on Matrix Distributions

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We resume results of the first author on classification of measurable functions in several variables, with some minor corrections of purely technical nature. We give a partial solution of the characterization problem for so-called matrix distributions which are metric invariants of measurable functions introduced by the first author. Matrix distributions are considered as (Sℕ × Sℕ)-invariant, ergodic measures on the space of matrices; this fact connects our problem with Aldous’ and Hoover’s theorem. Bibliography: 14 titles.

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References

  1. D. J. Aldous, “Exchangeability and related topics,” Lect. Notes Math., 1117, Springer (2006).

  2. D. J. Aldous, “Representations for partially exchangeable arrays of random variables,” J. Multivariate Analysis, 11, 581–598 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  3. V. I. Bogachev, Measure Theory, Vol. I, Springer, Berlin–Heidelberg (2007).

    Book  MATH  Google Scholar 

  4. M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics, 152, Birkhäuser Boston, Boston, MA (1999).

  5. U. Haböck, “Cohomology and Classifications Problems in Dynamics,” PhD Thesis, Faculty of Mathematics, Univ. of Vienna (2006).

  6. D. N. Hoover, “Row-column exchangeability and a generalized model for exchangeability,” in: G. Koch and F. Spizzichino (eds), Exchangeability in Probability and Statistics, North–Holland, Amsterdam (1982), pp. 281–291.

  7. V. A. Rokhlin, “Metric classification of measurable functions,” Usp. Mat. Nauk, 12, 169–174 (1957).

    MathSciNet  Google Scholar 

  8. A. M. Vershik, “Invariant measures – new aspects of dynamics, combinatorics, and representation theory,” Takagi Lectures, Math. Inst. of Tōhoku University (2015).

  9. A. M. Vershik, “On classification of measurable functions of several variables,” Zap. Nachn. Semin. POMI, 403, 35–57 (2012).

    MathSciNet  Google Scholar 

  10. A. M. Vershik and U. Haböck, “Compactness of the congruence group of measurable functions in several variables,” J. Math. Sciences, 141, 1601–1607 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  11. A. M. Vershik, “Random metric spaces and universality,” Russian Math. Surveys, 59, 259–295 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. M. Vershik, “Classification of measurable functions of several arguments, and invariantly distributed random matrices,” Funkt. Anal. Prilozh., 36, 12–27 (2002).

    Article  MathSciNet  Google Scholar 

  13. A. M. Vershik, “A random metric space is a Urysohn space,” Dokl. Ros. Akad. Nauk, 387, 733–736 (2002).

    MathSciNet  MATH  Google Scholar 

  14. A. M. Vershik, “Description of invariant measures for the actions of some infinite–dimensional groups,” Dokl. Akad. Nauk SSSR, 218, 749–752 (1974).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg State University, St.Petersburg, Russia

    A. M. Vershik

  2. Competence Centre for IT–Security, Wien, Austria

    U. Haböck

Authors
  1. A. M. Vershik
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  2. U. Haböck
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Correspondence to A. M. Vershik.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 441, 2015, pp. 119–143.

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Vershik, A.M., Haböck, U. On the Classification Problem of Measurable Functions in Several Variables and on Matrix Distributions. J Math Sci 219, 683–699 (2016). https://doi.org/10.1007/s10958-016-3138-x

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  • DOI: https://doi.org/10.1007/s10958-016-3138-x

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