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

This article is cited in 9 scientific papers (total in 9 papers)

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


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

Bibliographic databases:

UDC: 517.518.2+517.517
Received: 31.05.2007

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
\by V.~K.~Zakharov, T.~V.~Rodionov
\paper A Class of Uniform Functions and Its Relationship with the Class of Measurable Functions
\jour Mat. Zametki
\yr 2008
\vol 84
\issue 6
\pages 809--824
\jour Math. Notes
\yr 2008
\vol 84
\issue 6
\pages 756--770

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    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Zakharov V.K., Mikhalev A.V., Rodionov T.V., “Characterization of general Radon integrals”, Dokl. Math., 82:1 (2010), 613–616  crossref  mathscinet  zmath  isi  elib  elib  scopus
    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  mathnet  crossref  mathscinet
    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  mathnet  crossref
    5. V. K. Zakharov, T. V. Rodionov, “Naturalness of the Class of Lebesgue–Borel–Hausdorff Measurable Functions”, Math. Notes, 95:4 (2014), 500–508  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathnet  crossref  mathscinet
    7. Rodionov T.V., Zakharov V.K., “A Fine Correlation Between Baire and Borel Functional Hierarchies”, Acta Math. Hung., 142:2 (2014), 384–402  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref  mathscinet
    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  crossref  mathscinet  zmath  isi  elib  scopus
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