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Sigma-compactness of metric boolean algebras and uniform convergence of frequencies to probabilities

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Abstract

The topological properties of a pseudometric space defined by a measure are investigated. Criteria of compactness and σ-compactness of this space are proved. A new sufficient condition for the uniform convergence (over an event class) of frequencies to probabilities is proved as a corollary.

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References

  1. V. N. Vapnik and A. Ya. Chervonenkis, Pattern Recognition Theory (Nauka, Moscow, 1974) [in Russian].

    Google Scholar 

  2. V. N. Vapnik, Statistical Learning Theory (Wiley, New York, 1998).

    MATH  Google Scholar 

  3. N. Dunford and J. Schwartz, Linear Operators: General Theory (Interscience, New York, 1958; Inostrannaya Literatura, Moscow, 1962).

    MATH  Google Scholar 

  4. G. Birkhoff, Lattice Theory (Amer. Math. Soc., New York, 1940; Nauka, Moscow, 1984).

    Google Scholar 

  5. V. M. Kadets, A Course of Functional Analysis (Izd. Khar’kovsk. Nats. Univ., Kharkiv, 2004) [in Russian].

    Google Scholar 

  6. A. A. Borovkov, Mathematical Statistics (Nauka, Novosibirsk, 1997; Gordon & Breach, Amsterdam, 1998).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Division of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    E. G. Pytkeev & M. Yu. Khachai

Authors
  1. E. G. Pytkeev
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  2. M. Yu. Khachai
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Corresponding author

Correspondence to E. G. Pytkeev.

Additional information

Original Russian Text © E.G. Pytkeev, M.Yu.Khachai, 2010, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2010, Vol. 16, No. 1.

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Pytkeev, E.G., Khachai, M.Y. Sigma-compactness of metric boolean algebras and uniform convergence of frequencies to probabilities. Proc. Steklov Inst. Math. 272 (Suppl 1), 138–151 (2011). https://doi.org/10.1134/S0081543811020118

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  • DOI: https://doi.org/10.1134/S0081543811020118

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