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Mat. Sb. (N.S.), 1986, Volume 131(173), Number 1(9), Pages 73–93 (Mi msb1905)  

This article is cited in 18 scientific papers (total in 18 papers)

On the question of universal integrability of bounded functions

A. G. Chentsov


Abstract: Various procedures are considered for integration of bounded real-valued functions with respect to finitely additive measures on a semi-algebra of sets. A criterion is established for the indefinite Darboux integrals of a bounded function to coincide on the family of all positive finitely additive measures: a bounded function is universally integrable if and only if it belongs to the closure, in the metric of uniform convergence, of the linear span of the family of all characteristic functions of sets in the semi-algebra. A representation of the indefinite Darboux integral is obtained for such bounded functions. For arbitrary bounded functions a construction is proposed for a multivalued indefinite integral with respect to a positive finitely additive measure, and some of the properties of the construction are established. In particular, the multivalued integral of an arbitrary bounded function with respect to a positive countably additive measure consists only of countably additive measures of bounded variation, while the multivalued integral with respect to a purely finitely additive positive measure consists only of purely finitely additive measures. The dependence of the multivalued integral on the bounded function is continuous in the sense of the natural metric for the space of nonempty order intervals in the family of finitely additive measures of bounded variation.
Bibliography: 7 titles.

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English version:
Mathematics of the USSR-Sbornik, 1988, 59:1, 75–94

Bibliographic databases:

UDC: 517.987
MSC: Primary 28A25, 28A33; Secondary 28B20
Received: 02.08.1984 and 03.09.1985

Citation: A. G. Chentsov, “On the question of universal integrability of bounded functions”, Mat. Sb. (N.S.), 131(173):1(9) (1986), 73–93; Math. USSR-Sb., 59:1 (1988), 75–94

Citation in format AMSBIB
\Bibitem{Che86}
\by A.~G.~Chentsov
\paper On the question of universal integrability of bounded functions
\jour Mat. Sb. (N.S.)
\yr 1986
\vol 131(173)
\issue 1(9)
\pages 73--93
\mathnet{http://mi.mathnet.ru/msb1905}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=868602}
\zmath{https://zbmath.org/?q=an:0628.28004|0621.28004}
\transl
\jour Math. USSR-Sb.
\yr 1988
\vol 59
\issue 1
\pages 75--94
\crossref{https://doi.org/10.1070/SM1988v059n01ABEH003125}


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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. Chentsov A., “Finitely Additive Measures and Problems for Minimum”, 24, no. 3, 1988, 353–357  mathscinet  zmath  isi
    2. Chentsov A., “Finitely Additive Measures and Extensions of Some Nonlinear Extremal Problems with Asymptotic Constraints”, 26, no. 6, 1990, 902–911  mathscinet  zmath  isi
    3. Belov E., Chentsov A., “Extension and Stability of Some Multicriterion Problems”, 26, no. 5, 1990, 678–686  mathscinet  zmath  isi
    4. Serov V., Chentsov A., “Construction of an Extension of a Control Problem with Integral Restrictions”, Differ. Equ., 26:4 (1990), 442–450  mathnet  mathscinet  zmath  isi
    5. Chentsov A., “Asymptotic Effectiveness and Expansions in Class of Finite-Additive Measures”, 314, no. 5, 1990, 1085–1087  mathscinet  isi
    6. Chentsov A., “On the Construction of Solution to Nonregular Problems of Optimal-Control”, 20, no. 2, 1991, 129–143  mathscinet  zmath  isi
    7. A. G. Chentsov, “Relaksatsiya nekotorykh ekstremalnykh zadach s integralnymi ogranicheniyami”, Vestnik ChelGU, 1991, no. 1, 37–46  mathnet
    8. Chentsov A., “Relaxation of Reachable Sets and Extension Constructions”, Cybern. Syst. Anal., 28:4 (1992), 554–561  crossref  mathscinet  zmath  isi
    9. Chentsov A., “The Asymptotic Attainability and its Generalized Representation”, Dokl. Akad. Nauk, 334:4 (1994), 437–440  mathnet  mathscinet  zmath  isi
    10. S. I. Morina, A. G. Chentsov, “Ob odnoi zadache asimptoticheskoi optimizatsii”, Vestnik ChelGU, 1994, no. 2, 80–86  mathnet
    11. A. G. Chentsov, “Asimptotika dostizhimykh mnozhestv i obobschennye konstruktsii v klasse konechno-additivnykh mer”, Vestnik ChelGU, 1994, no. 2, 126–134  mathnet
    12. A. G. Chentsov, “On the correct extension of a problem of selecting the probability density under constraints on a system of mathematical expectations”, Russian Math. Surveys, 50:5 (1995), 1065–1084  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    13. Chentsov A., “Asymptotic Reachability with Perturbed Integral Constraints”, Cybern. Syst. Anal., 31:1 (1995), 75–84  crossref  mathscinet  zmath  isi
    14. Morina S., Chentsov A., “Bounded Realization of Asymptotically Attainable Elements”, Pmm-J. Appl. Math. Mech., 59:6 (1995), 951–956  crossref  mathscinet  zmath  isi
    15. L. A. Savinova, A. G. Chentsov, “K voprosu o rasshirenii nekotorykh volterrovykh operatorov v klasse konechno-additivnykh mer”, Vestnik ChelGU, 1996, no. 3, 117–134  mathnet
    16. V. I. Kuznetsov, A. G. Chentsov, “On a construction of an extension of an abstract, purely pulse control problem”, Comput. Math. Math. Phys., 37:5 (1997), 510–518  mathnet  mathscinet  zmath
    17. Berdyshev Y., Chentsov A., “Equivalence of Regularizations in Abstract Problems with Different Classes of Admissible Controls”, Cybern. Syst. Anal., 34:3 (1998), 377–385  crossref  mathscinet  zmath  isi
    18. Morina S., “On Approximation of Asymptotic Attainability Domains”, Nonsmooth and Discontinuous Problems of Control and Optimization (NDPCO'98), eds. Batukhtin V., Kirillova F., Ukhobotov V., Elsevier Science BV, 1999, 165–170  mathscinet  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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