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 Mat. Zametki, 2008, Volume 84, Issue 6, Pages 809–824 (Mi mz3998)

A Class of Uniform Functions and Its Relationship with the Class of Measurable Functions

V. K. Zakharova, T. V. Rodionovb

a Centre for New Information Technologies, Moscow State University
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Borel, Lebesgue, and Hausdorff described all uniformly closed families of real-valued functions on a set $T$ whose algebraic properties are just like those of the set of all continuous functions with respect to some open topology on $T$. These families turn out to be exactly the families of all functions measurable with respect to some $\sigma$-additive and multiplicative ensembles on $T$. The problem of describing all uniformly closed families of bounded functions whose algebraic properties are just like those of the set of all continuous bounded functions remained unsolved. In the paper, a solution of this problem is given with the help of a new class of functions that are uniform with respect to some multiplicative families of finite coverings on $T$. It is proved that the class of uniform functions differs from the class of measurable functions.

Keywords: uniform function, measurable function, measurable function w.r.t an ensemble, $\sigma$-additive ensemble, normal family of functions, boundedly normal family of functions

DOI: https://doi.org/10.4213/mzm3998

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English version:
Mathematical Notes, 2008, 84:6, 756–770

Bibliographic databases:

UDC: 517.518.2+517.517

Citation: V. K. Zakharov, T. V. Rodionov, “A Class of Uniform Functions and Its Relationship with the Class of Measurable Functions”, Mat. Zametki, 84:6 (2008), 809–824; Math. Notes, 84:6 (2008), 756–770

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz3998
• https://doi.org/10.4213/mzm3998
• http://mi.mathnet.ru/eng/mz/v84/i6/p809

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “The Riesz–Radon–Fréchet problem of characterization of integrals”, Russian Math. Surveys, 65:4 (2010), 741–765
2. Zakharov V.K., Mikhalev A.V., Rodionov T.V., “Characterization of general Radon integrals”, Dokl. Math., 82:1 (2010), 613–616
3. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Characterization of Radon integrals as linear functionals”, J. Math. Sci., 185:2 (2012), 233–281
4. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “The characterization of integrals with respect to arbitrary Radon measures by the boundedness indices”, J. Math. Sci., 185:3 (2012), 417–429
5. V. K. Zakharov, T. V. Rodionov, “Naturalness of the Class of Lebesgue–Borel–Hausdorff Measurable Functions”, Math. Notes, 95:4 (2014), 500–508
6. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Descriptive spaces and proper classes of functions”, J. Math. Sci., 213:2 (2016), 163–200
7. Rodionov T.V., Zakharov V.K., “A Fine Correlation Between Baire and Borel Functional Hierarchies”, Acta Math. Hung., 142:2 (2014), 384–402
8. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Postclassical families of functions proper for descriptive and prescriptive spaces”, J. Math. Sci., 221:3 (2017), 360–383
9. Zakharov V.K., Mikhalev A.V., Rodionov T.V., “Postclassical Families of Functions Proper To Descriptive and Prescriptive Spaces”, Dokl. Math., 92:2 (2015), 559–562
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