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 Trudy Inst. Mat. i Mekh. UrO RAN, 2013, Volume 19, Number 2, Pages 307–319 (Mi timm956)

On the question of representation of ultrafilters in a product of measurable spaces

A. G. Chentsovab

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
b Ural Federal University

Abstract: We study representations of ultrafilters of widely understood measurable spaces that are realized by means of generalized Cartesian products. The structure of the arising ultrafilter space is established; in more traditional measurable spaces, this structure reduces to the realization of the Stone compact space in the form of a Tikhonov product. The methods developed can be applied to the construction of extensions of abstract attainability problems with constraints of asymptotic nature; in such extensions, ultrafilters can be used as generalized elements, which admits a conceptual analogy with the Stone–Cech compactification. The proposed implementation includes the possibility of using measurable spaces, for which the set of free ultrafilters can be described completely. This results in an exhaustive representation of the corresponding Stone compact space for measurable spaces with algebras of sets. The present issue is devoted to I. I. Eremin's jubilee; the author had many discussion with him on very different topics related to mathematical investigations, and the discussions inevitably led to a deeper understanding of their essence. The author appreciates the possibility of such communication and is grateful to Eremin, who contributed significantly to the development of the mathematical science and education in the Urals.

Keywords: measurable space, Tychonoff product, ultrafilter.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2014, 284, suppl. 1, 65–78

Bibliographic databases:

Document Type: Article
UDC: 519.6

Citation: A. G. Chentsov, “On the question of representation of ultrafilters in a product of measurable spaces”, Trudy Inst. Mat. i Mekh. UrO RAN, 19, no. 2, 2013, 307–319; Proc. Steklov Inst. Math. (Suppl.), 284, suppl. 1 (2014), 65–78

Citation in format AMSBIB
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This publication is cited in the following articles:
1. A. G. Chentsov, “On the question of representation of ultrafilters and their application in extension constructions”, Proc. Steklov Inst. Math. (Suppl.), 287, suppl. 1 (2014), 29–48
2. A. G. Chentsov, “K voprosu o predstavlenii kompaktov Stouna”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 4, 156–174
3. A. G. Chentsov, “Ultrafiltry izmerimykh prostranstv i ikh primenenie v konstruktsiyakh rasshirenii”, Tr. IMM UrO RAN, 20, no. 1, 2014, 285–304
4. E. G. Pytkeev, A. G. Chentsov, “On the structure of ultrafilters and properties related to convergence in topological spaces”, Proc. Steklov Inst. Math. (Suppl.), 289, suppl. 1 (2015), 164–181
5. A. G. Chentsov, A. P. Baklanov, I. I. Savenkov, “Zadacha o dostizhimosti s ogranicheniyami asimptoticheskogo kharaktera”, Izv. IMI UdGU, 2016, no. 1(47), 54–118
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