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## Archive of Issues

Russia Yekaterinburg
Year
2016
Volume
26
Issue
3
Pages
345-365
 Section Mathematics Title Some representations of free ultrafilters Author(-s) Pytkeev E.G.ab, Chentsov A.G.ab Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Ural Federal Universityb Abstract Constructions related to the representation of free $\sigma$-multiplicative ultrafilters of widely interpreted measurable spaces are considered. These constructions are based on the representations connected with the application of open ultrafilters for co-finite and co-countable topologies. Such ultrafilters are preserved (as maximal filters) under the replacement of topologies by algebra and $\sigma$-algebra generated by above-mentioned topologies, respectively. In (general) case of co-countable topology, uniqueness of $\sigma$-multiplicative free ultrafilter composed of nonempty open sets is established. It is demonstrated that the given property is preserved for $\sigma$-algebras containing co-countable topology. Two topologies of the space of bounded finitely additive Borel measures with the property of uniqueness of remainder for sequentially closed set of Dirac measures under the closure construction are stated. Keywords algebra of sets, measure, topology, ultrafilter UDC 519.6 MSC 28A33 DOI 10.20537/vm160305 Received 1 July 2016 Language Russian Citation Pytkeev E.G., Chentsov A.G. Some representations of free ultrafilters, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 3, pp. 345-365. References Engelking R. Obshchaya topologiya (General topology), Moscow: Mir, 1986, 751 p. Chentsov A.G. On measurable spaces admitting non-Dirac countably additive $(0,1)$-measure, Doklady Mathematics, 2002, vol. 65, no. 3, pp. 425-428. Chentsov A.G. Structure of countably additive non-Dirac $(0,1)$-measures, Matematicheskii i prikladnoi analiz: sbornik nauchnykh trudov, vyp. 1 (Mathematical and applied analysis: Transactions, issue 1), Tyumen: Tyumen State University, 2003, pp. 218-243 (in Russian). Chentsov A.G. Elementy konechno-additivnoi teorii mery, II (Elements of a finitely additive measure theory, II), Yekaterinburg: USTU-UPI, 2010, 542 p. Iliadis S., Fomin S. The method of centred systems in the theory of topological spaces, Russian Mathematical Surveys, 1966, vol. 21, no. 4, pp. 37-62. DOI: 10.1070/RM1966v021n04ABEH004165 Bulinskii A.V., Shiryaev A.N. Teoriya sluchainykh protsessov (Theory of stochastic processes), Moscow: Fizmatlit, 2005, 402 p. Chentsov A.G., Morina S.I. Extensions and relaxations, Dordrecht-Boston-London: Kluwer Academic Publishers, 2002, 408 p. DOI: 10.1007/978-94-017-1527-0 Chentsov A.G., Pytkeev E.G. Some topological structures of extensions of abstract reachability problems, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 292, suppl. 1, pp. 36-54. DOI: 10.1134/S0081543816020048 Pytkeev E.G., Chentsov A.G. Some properties of open ultrafilters, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2015, no. 2 (46), pp. 140-148 (in Russian). http://mi.mathnet.ru/iimi314 Burbaki N. Obshchaya topologiya (General topology), Moscow: Nauka, 1968, 272 p. Aleksandryan R.A., Mirzakhanyan E.A. Obshchaya topologiya (General topology), Moscow: Vysshaya shkola, 1979, 336 p. Chentsov A.G. To question about realization of attraction elements in abstract attainability problems, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2015, vol. 25, no. 2, pp. 212-229 (in Russian). DOI: 10.20537/vm150206 Chentsov A.G. Elementy konechno-additivnoi teorii mery, I (Elements of a finitely additive measure theory, II), Yekaterinburg: USTU-UPI, 2009, 389 p. Chentsov A.G. To question about representation of Stone compactums, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 4, pp. 156-174 (in Russian). DOI: 10.20537/vm130415 Full text