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

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

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
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
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
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
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
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
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
8. V. K. Zakharov, T. V. Rodionov, “Classification of Borel sets and functions for an arbitrary space”, Sb. Math., 199:6 (2008), 833–869
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
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
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
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