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Izv. Akad. Nauk SSSR Ser. Mat., 1990, Volume 54, Issue 5, Pages 928–956 (Mi izv1056)  

This article is cited in 11 scientific papers (total in 11 papers)

Connections between the Lebesgue extension and the Borel extension of the first class, and between the preimages corresponding to them

V. K. Zakharov


Abstract: A new algebraic structure of a $c$-ring with refinement and a new topological structure of an $a$-space with cover are introduced. On the basis of them the notions of divisible hulls and surrounded coverings of certain types are introduced. With the help of these notions the Lebesgue extension $C\rightarrowtail L_\mu$ and the Borel extension $C\rightarrowtail BM_1$ of the first class are given a ring characterization as divisible hulls of a certain type (Theorem 1); preimages of maximal ideals of these extensions are given a topological characterization as surrounded coverings of a certain type (Theorem 2).

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English version:
Mathematics of the USSR-Izvestiya, 1991, 37:2, 273–302

Bibliographic databases:

UDC: 512.552+515.12
MSC: Primary 54C40, 46J10; Secondary 54C50
Received: 28.01.1988

Citation: V. K. Zakharov, “Connections between the Lebesgue extension and the Borel extension of the first class, and between the preimages corresponding to them”, Izv. Akad. Nauk SSSR Ser. Mat., 54:5 (1990), 928–956; Math. USSR-Izv., 37:2 (1991), 273–302

Citation in format AMSBIB
\Bibitem{Zak90}
\by V.~K.~Zakharov
\paper Connections between the Lebesgue extension and the Borel extension of the first class, and between the preimages corresponding to them
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1990
\vol 54
\issue 5
\pages 928--956
\mathnet{http://mi.mathnet.ru/izv1056}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1086080}
\zmath{https://zbmath.org/?q=an:0771.54014}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1991IzMat..37..273Z}
\transl
\jour Math. USSR-Izv.
\yr 1991
\vol 37
\issue 2
\pages 273--302
\crossref{https://doi.org/10.1070/IM1991v037n02ABEH002064}


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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, “The regular and the Baire extension of the ring of continuous functions as rings of quotients of the same type”, Russian Math. Surveys, 46:6 (1991), 235–236  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. V. K. Zakharov, “The Kaplan extension of the ring and Banach algebra of continuous functions as a divisible hull”, Russian Acad. Sci. Izv. Math., 45:3 (1995), 477–493  mathnet  crossref  mathscinet  zmath  isi
    3. V. K. Zakharov, “Extensions of the ring of continuous functions generated by the classical, rational, and regular rings of fractions as divisible hulls”, Sb. Math., 186:12 (1995), 1773–1809  mathnet  crossref  mathscinet  zmath  isi
    4. V. K. Zakharov, “Svyaz mezhdu klassicheskim koltsom chastnykh koltsa nepreryvnykh funktsii i funktsiyami, integriruemymi po Rimanu”, Fundament. i prikl. matem., 1:1 (1995), 161–176  mathnet  mathscinet  zmath  elib
    5. V. K. Zakharov, “Extensions of the ring of continuous functions generated by regular, countably-divisible, complete rings of quotients, and their corresponding pre-images”, Izv. Math., 59:4 (1995), 677–720  mathnet  crossref  mathscinet  zmath  isi
    6. V. K. Zakharov, A. V. Mikhalev, “The problem of general Radon representation for an arbitrary Hausdorff space”, Izv. Math., 63:5 (1999), 881–921  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. V. K. Zakharov, A. V. Mikhalev, “The problem of general Radon representation for an arbitrary Hausdorff space. II”, Izv. Math., 66:6 (2002), 1087–1101  mathnet  crossref  crossref  mathscinet  zmath
    8. V. K. Zakharov, T. V. Rodionov, “Classification of Borel sets and functions for an arbitrary space”, Sb. Math., 199:6 (2008), 833–869  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. 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
    10. 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
    11. 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
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