Abstract
Consideration is given to representations of limiting process operations in terms of ultrafilters of measurable spaces and in terms of (0,1)-measures generated by these ultrafilters. Properties that are specified by the existence of non-Dirac countably additive (0,1)-measures and have the sense of degeneracy of corresponding “limiting” operations are studied. Properties of spaces whose elements are uniform limits of step mappings called tier mappings are analysed. We consider applications of these mappings in constructions connected with the design of correct extensions for abstract problems of attainability that admit natural analogs with problems of construction and study of properties of attainability domains of controlled systems.
Similar content being viewed by others
References
Neveu, J., Bases mathématiques du calcul des probabilités, Paris: Masson, 1964. Translated under the title Matematicheskie osnovy teorii veroyatnostei, Moscow: Mir, 1969.
Pkhakadze, Sh.S., Measure Decomposition, Tr. Tbilis. Mat. Inst., 1964, vol. XXIX, pp. 121–145.
Chentsov, A.G., On One Class of non-Dirac Countably Additive (0,1)-measures, in Funktsional’nodifferentsial’nye uravneniya. Vest. Perm. Gos. Tekh. Univ. (spetsial’nyi vypusk) (Functional Differential Equations. Bull. Perm. Tech. Gos. Univ. (Special Issue)), 2002, pp. 238–252.
Arkhangel’skii, A.V., Topologicheskie prostranstva funktsii (Topological Spaces of Functions), Moscow: Mosk. Gos. Univ., 1989.
Belov, E.G. and Chentsov, A.G., Certain Properties of Two-Valued Measures and Conditions of Universal Integrability, Mat. Zametki, 1987, vol. 42, no. 2, pp. 288–297.
Chentsov, A.G., Certain Properties of Two-Valued Measures and Representations of Filter Limit, Dokl. Akad. Nauk, 2000, vol. 370, no. 5, pp. 595–598.
Chentsov, A.G., Representation of Sets of Countably Additive Non-Dirac (0,1)-measures, Dokl. Akad. Nauk, 2001, vol. 377, no. 3, pp. 313–316.
Chentsov, A.G., Non-Dirac (0,1)-measures and σ-topological Spaces, Vestn. Chelyab. Univ., 2003, no. 2(8), pp. 190–202.
Chentsov, A.G., Structure of Countably Additive Non-Dirac (0,1)-measures, Mat. Prikl. Analiz., 2003, no. 1, pp. 218–243.
Chentsov, A.G., Two-Valued Measures: Finite Additivity and Countable Additivity, Functional Diff. Equat., 2000, no. 3–4, pp. 231–257.
Kelley, J.L., General Topology, New York: Van Nostrand, 1957. Translated under the title Obshchaya topologiya, Moscow: Nauka, 1981.
Engelking, R., General Topology, Warzawa: PWN, 1985. Translated under the title Obshchaya topologiya, Moscow: Mir, 1986.
Alexandroff, A.D., Additive Set-Function in Abstract Spaces, Mat. Sb., 1940, vol. 8(50), no. 2, pp. 307–348.
Erokhin, V.D., Notes about Measure Theory, Uspekhi Mat. Nauk, 1961, vol. XVI, no. 3(9), pp. 175–180.
Chentsov, A.G., Certain Problems of Asymptotic Analysis and Extensions. I, Avtom. Telemekh., 2005, no. 12, pp. 125–142.
Chentsov, A.G., Certain Problems of Asymptotic Analysis and Extensions. II, Avtom. Telemekh., 2006, no. 1, pp. 128–145.
Chentsov, A.G., Asymptotic Attainability, Dordrecht: Kluwer, 1997.
Chentsov, A.G. and Morina, S.I., Extensions and Relaxations, Dordrecht: Kluwer, 2002.
Kuratowski, K. and Mostowski, A., Set Theory, New York: North-Holland, 1976. Translated under the title Teoriya mnozhestv, Moscow: Mir, 1970.
Chentsov, A.G., Approximate Solutions in the Problem of Asymptotic Attainability, Izv. Vuzov, Mat., 2005, no. 8, pp. 63–73.
Melentsov, A.A., Baidosov, V.A., and Zmeev, G.M., Elementy teorii mery i integrala (Elements of the Theory of Measure and Integral), Sverdlovsk: Ural. Univ., 1980.
Dunford, N. and Schwartz, J.T., Linear Operators, New York: Interscience, 1958. Translated under the title Lineinye operatory. Obshchaya teoriya, Moscow: Inostrannaya Literatura, 1962.
Billingsley, P., Convergence of Probability Measures, New York: Wiley, 1968. Translated under the title Skhodimost’ veroyatnostnykh mer, Moscow: Nauka, 1977.
Chentsov, A.G., Certain Constructions of Asymptotic Analysis Related to Stone-Czech Compactification, Sovr. Mat. Prilozh., Akad. Nauk Gruzii, Inst. Kibern., 2005, vol. 26, pp. 119–150.
Chentsov, A.G., Nonsequential Approximate Solutions in Abstract Problems of Attainability, Tr. Inst. Mat. Mekh., Ural. Otdel. Ross. Akad. Nauk, Yekaterinburg, 2006, vol. 12, pp. 216–241.
Bourbaki, N., Topologie Générale, Paris: Hermann, 1958. Translated under the title Obshchaya topologiya, Moscow: Nauka, 1968.
Krasovskii, N.N., Teoriya upravleniya dvizheniem (Theory of Motion Control), Moscow: Nauka, 1968.
Krasovskii, N.N., Igrovye zadachi o vstreche dvizhenii (Game Problems of Motion Encounter), Moscow: Nauka, 1970.
Panasyuk, A.I. and Panasyuk, V.I., Asimptoticheskaya magistral’naya optimizatsiya upravlyaemykh sistem (Asymptotic Main Line Optimization of Controlled Systems), Minsk: Nauka i Tekhnika, 1986.
Chentsov, A.G., The Nonsequential Approximate Solutions in Problems of Asymptotic Analysis, Soochow J. Math., 2006, vol. 32, no. 3, pp. 441–475.
Warga, J., Optimal Control of Differential and Functional Equations, New York: Academic, 1972. Translated under the title Optimal’noe upravlenie differenstial’nymi i funktsional’nymi uravneniyami, Moscow: Nauka, 1977.
Chentsov, A.G., Nonsequential Approximate Solutions in Abstract Control Problems, Trudy Mezhdunarodnogo seminara “Teoriya upravleniya i teoriya obobshchennykh reshenii uravnenii Gamil’tona-Yakobi” (Proc. Int. Seminar “Control Theory and Theory of Generalized Solutions to Hamilton-Jacobi Equations”), Yekaterinburg: Ural. Univ., 2006, vol. 1, pp. 48–60.
Chentsov, A.G., Konechno-additivnye mery i relaksatsii ekstremal’nykh zadach, Yekaterinburg: Nauka, 1993. Translated under the title Finitely Additive Measures and Relaxations of Extremal Problems, New York: Plenum, 1996.
Author information
Authors and Affiliations
Additional information
Original Russian Text © A.G. Chentsov, 2007, published in Avtomatika i Telemekhanika, 2007, No. 11, pp. 208–222.
This work was supported by the Russian Foundation for Basic Research, project no. 06-01-00414.
Rights and permissions
About this article
Cite this article
Chentsov, A.G. Construction of limiting process operations using ultrafilters of measurable spaces. Autom Remote Control 68, 2083–2096 (2007). https://doi.org/10.1134/S000511790711015X
Received:
Issue Date:
DOI: https://doi.org/10.1134/S000511790711015X