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

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

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
2. Chentsov A., “Finitely Additive Measures and Extensions of Some Nonlinear Extremal Problems with Asymptotic Constraints”, 26, no. 6, 1990, 902–911
3. Belov E., Chentsov A., “Extension and Stability of Some Multicriterion Problems”, 26, no. 5, 1990, 678–686
4. Serov V., Chentsov A., “Construction of an Extension of a Control Problem with Integral Restrictions”, Differ. Equ., 26:4 (1990), 442–450
5. Chentsov A., “Asymptotic Effectiveness and Expansions in Class of Finite-Additive Measures”, 314, no. 5, 1990, 1085–1087
6. Chentsov A., “On the Construction of Solution to Nonregular Problems of Optimal-Control”, 20, no. 2, 1991, 129–143
7. A. G. Chentsov, “Relaksatsiya nekotorykh ekstremalnykh zadach s integralnymi ogranicheniyami”, Vestnik ChelGU, 1991, no. 1, 37–46
8. Chentsov A., “Relaxation of Reachable Sets and Extension Constructions”, Cybern. Syst. Anal., 28:4 (1992), 554–561
9. Chentsov A., “The Asymptotic Attainability and its Generalized Representation”, Dokl. Akad. Nauk, 334:4 (1994), 437–440
10. S. I. Morina, A. G. Chentsov, “Ob odnoi zadache asimptoticheskoi optimizatsii”, Vestnik ChelGU, 1994, no. 2, 80–86
11. A. G. Chentsov, “Asimptotika dostizhimykh mnozhestv i obobschennye konstruktsii v klasse konechno-additivnykh mer”, Vestnik ChelGU, 1994, no. 2, 126–134
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
13. Chentsov A., “Asymptotic Reachability with Perturbed Integral Constraints”, Cybern. Syst. Anal., 31:1 (1995), 75–84
14. Morina S., Chentsov A., “Bounded Realization of Asymptotically Attainable Elements”, Pmm-J. Appl. Math. Mech., 59:6 (1995), 951–956
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
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
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
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
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