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On a Choquet-Stieltjes type integral on intervals

  • Sorin G. Gal EMAIL logo
Published/Copyright: July 19, 2019
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Abstract

In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

MSC 2010: Primary 28A25; 26A42
  1. (Communicated by Anatolij Dvurečenskij)

References

[1] Agahi, H.—Mesiar, R.: Stolarsky’s inequality for Choquet-like expectation, Math. Slovaca 66(5) (2016), 1235–1248.10.1515/ms-2016-0219Search in Google Scholar

[2] Agahi, H.—Mesiar, R.: On Choquet-Pettis expectation of Banach-valued functiona : a counter example, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 26(2) (2018), 255–259.10.1142/S0218488518500137Search in Google Scholar

[3] Boccuto, A.—Riečan, B.: The symmetric Choquet integral with respect to Riesz-space-valued capacities, Czechoslovak Math. J. 58(2) (2008), 289–310.10.1007/s10587-008-0017-8Search in Google Scholar

[4] Boccuto, A.—Riečan, B.: The concave integral with respect to Riesz-space-valued capacities, Math. Slovaca 59(6) (2009), 647–660.10.2478/s12175-009-0153-0Search in Google Scholar

[5] Borel, E.: Modèles arithmétiques et analytiques de ľirréversibilité apparente, Compte Rendus Acad. Sci. Paris 154 (1912), 1148–1150.Search in Google Scholar

[6] Candeloro, D.—Mesiar, R.—Sambucini, A. R.: A special class of fuzzy measures : Choquet integral and applications, Fuzzy Sets and Systems 355 (2019), 83–99.10.1016/j.fss.2018.04.008Search in Google Scholar

[7] Choquet, G.: Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1954), 131–292.10.5802/aif.53Search in Google Scholar

[8] Denneberg, D.: Non-Additive Measure and Integral, Kluwer Academic Publisher, Dordrecht, Boston, London, 2010.Search in Google Scholar

[9] Doria, S.—Dutta, B.—Mesiar, R.: Integral representation of coherent upper conditional prevision with respect to its associated Hausdorff outer measure : a comparison among the Choquet integral, the pan-integral and the concave integral, Int. J. Gen. Syst. 47(6) (2018), 569–592.10.1080/03081079.2018.1473392Search in Google Scholar

[10] Dubois, D.—Prade, H.: Possibility Theory, Plenum Press, New York, 1988.10.1007/978-1-4684-5287-7Search in Google Scholar

[11] Gal, S. G.: Approximation by nonlinear Choquet integral operators, Ann. Mat. Pura Appl. 195(3) (2016), 881–896.10.1007/s10231-015-0495-xSearch in Google Scholar

[12] Gal, S. G. : Uniform and pointwise quantitative approximation by Kantorovich-Choquet type integral operators with respect to monotone and submodular set functions, Mediterr. J. Math. 14(5) (2017), 205–216.10.1007/s00009-017-1007-6Search in Google Scholar

[13] Gal, S. G. : The Choquet integral in capacity, Real Anal. Exchange 43(2) (2018), 263–280.10.14321/realanalexch.43.2.0263Search in Google Scholar

[14] Gal, S. G.—Opris, B. D.: Uniform and pointwise convergence of Bernstein-Durrmeyer operators with respect to monotone and submodular set functions, J. Math. Anal. Appl. 424 (2015), 1374–1379.10.1016/j.jmaa.2014.12.012Search in Google Scholar

[15] Gal, S. G.—Trifa, S.: Quantitative estimates in uniform and pointwise approximation by Bernstein-Durrmeyer-Choquet operators, Carpathian J. Math. 33 (2017), 49–58.10.37193/CJM.2017.01.06Search in Google Scholar

[16] Gal, S. G. : Fredholm-Choquet integral equations, J. Integral Equations Appl., to appear, https://projecteuclid.org/euclid.jiea/1542358961.10.1216/JIE-2019-31-2-183Search in Google Scholar

[17] Gal, S. G. : Volterra-Choquet integral equations, J. Integral Equations Appl., to appear, https://projecteuclid.org/euclid.jiea/1541668067.10.1216/JIE-2019-31-4-495Search in Google Scholar

[18] Gal, S. G.: Quantitative approximation by Stancu-Durrmeyer-Choquet-Šipoš operators, Math. Slovaca, accepted for publication.10.1515/ms-2017-0252Search in Google Scholar

[19] Goubault-Larrecq, J.: Une introduction aux capacités, aux jeux et aux prévisions, LSV/ UMR 8643 CNRS - ENS Cachan, INRIA Futurs projet SECSI 61, Version 1.8 préliminaire du 28 juin 2007.Search in Google Scholar

[20] Hincin, A. I.: Researches sur la structure des fonctions measurables, Fund. Math. 9 (1927), 212–279.10.4064/fm-9-1-212-279Search in Google Scholar

[21] Iyengar, A.: Analysis Constructing a Sequence, (https://math.stackexchange.com/users/95668/ashwin-iyengar), (version: 2015-05-12): https://math.stackexchange.com/q/1276535.Search in Google Scholar

[22] Mesiar, R.: Choquet-like integrals, J. Math. Anal. Appl. 194(2) (1995), 477–488.10.1006/jmaa.1995.1312Search in Google Scholar

[23] Mesiar, R.—Li, J.—Pap, E.: The Choquet integral as Lebesgue integral and related inequalities, Kybernetika (Prague) 46(6) (2010), 1098–1107.Search in Google Scholar

[24] Mihailovič, B.—Pap, E.: Asymmetric integral as a limit of generated Choquet integrals based on absolutely monotone real set functions, Fuzzy Sets and Systems 181 (2011), 39–49.10.1016/j.fss.2011.05.007Search in Google Scholar

[25] Narukawa, Y.—Murofushi, T.: Decision modelling using the Choquet integral. In: Internat. Conf. on Modelling Decision Artificial Intelligence. MDAI 2004, Barcelona, 2004, pp. 183–193.10.1007/978-3-540-27774-3_18Search in Google Scholar

[26] Narukawa, Y.—Murofushi, T.: Choquet-Stieltjes integral as a tool for decision modelling, Int. J. Intell. Syst. 23 (2008), 115–127.10.1002/int.20260Search in Google Scholar

[27] Narukawa, Y.—Torra, V.—Sugeno, M.: Choquet integral with respect to a symmetric fuzzy measure of a function on the real line, Ann. Oper. Res. 244(2) (2016), 571–581.10.1007/s10479-012-1166-6Search in Google Scholar

[28] Ouyang, Y.—LI, J.—Mesiar, R.: On the equivalence of the Choquet, pan and concave integrals on finite spaces, J. Math. Anal. Appl. 456(1) (2017), 151–162.10.1016/j.jmaa.2017.06.086Search in Google Scholar

[29] Ridaoui, M.—Grabisch, M.: Choquet integral calculus on a continuous support and its applications, Oper. Res. Decis. 26(1) (2016), 73–93.Search in Google Scholar

[30] Sambucini, A. R.: The Choquet integral with respect to fuzzy measures and applications, Math. Slovaca 67(6) (2017), 1427–1450.10.1515/ms-2017-0049Search in Google Scholar

[31] Sugeno, M.: Theory of Fuzzy Integrals and its Applications, Ph.D. dissertation. Tokyo Institute of Technology, Tokyo, 1974.Search in Google Scholar

[32] Sugeno, M.: A way to Choquet calculus, IEEE Trans. Fuzzy Systems 23(5) (2015), 1439–1457.10.1109/TFUZZ.2014.2362148Search in Google Scholar

[33] Vitali, G.: On the definition of integral of functions of one variable, Rivista di matematica per le scienze economiche e sociali. 20(2) (1925), 159–168.10.1007/BF02728999Search in Google Scholar

[34] Wang, S.—Klir, G. J.: Generalized Measure Theory, Springer, New York, 2009.10.1007/978-0-387-76852-6Search in Google Scholar

Received: 2018-12-02
Accepted: 2018-12-28
Published Online: 2019-07-19
Published in Print: 2019-08-27

© 2019 Mathematical Institute Slovak Academy of Sciences

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