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 Fundam. Prikl. Mat., 2014, Volume 19, Issue 2, Pages 51–107 (Mi fpm1578)

Descriptive spaces and proper classes of functions

V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov

Lomonosov Moscow State University, Moscow, Russia

Abstract: The remarkable class of measurable functions was introduced by classics of function theory. It has found different applications in various branches of mathematics. However this class turned out too restrictive for solving some natural mathematical problems because it is essentially connected with the property of countability. Therefore, along with it another remarkable class, essentially connected with the property of finiteness, was introduced. It is the class of uniform functions. Measurable functions are described both in the classical Lebesgue–Borel language of preimages and in the quite new language of covers. Uniform functions are described in the language of covers exclusively. Both the families of measurable functions and the families of uniform functions are determined by the rigid structure of their supports (descriptive spaces). For this reason mathematicians weakened more that once the rigidity of the structure of descriptive spaces at the expense of using the additional property of negligence. The present paper is devoted to a contemporary formalization of the indicated ideas. Some applications of the introduced classes of functions to solving a number of known mathematical problems is traced in the paper.

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English version:
Journal of Mathematical Sciences (New York), 2016, 213:2, 163–200

Bibliographic databases:

Document Type: Article
UDC: 517.517+517.518.2+510.225+515.128

Citation: V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Descriptive spaces and proper classes of functions”, Fundam. Prikl. Mat., 19:2 (2014), 51–107; J. Math. Sci., 213:2 (2016), 163–200

Citation in format AMSBIB
\Bibitem{ZakMikRod14} \by V.~K.~Zakharov, A.~V.~Mikhalev, T.~V.~Rodionov \paper Descriptive spaces and proper classes of functions \jour Fundam. Prikl. Mat. \yr 2014 \vol 19 \issue 2 \pages 51--107 \mathnet{http://mi.mathnet.ru/fpm1578} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3431916} \transl \jour J. Math. Sci. \yr 2016 \vol 213 \issue 2 \pages 163--200 \crossref{https://doi.org/10.1007/s10958-016-2709-1} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84954557984}