Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems / Chentsov A. G. // PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS. - 2011. - V. 275, l. 1. - P. 12-39.

ISSN/EISSN:

0081-5438 / нет данных

Type:

Article

Abstract:

We consider problems of asymptotic analysis that arise, in particular, in the formalization of effects related to an approximate observation of constraints. We study nonsequential (generally speaking) variants of asymptotic behavior that can be formalized in the class of ultrafilters of an appropriate measurable space. We construct attraction sets in a topological space that are realized in the class of ultrafilters of the corresponding measurable space and specify conditions under which ultrafilters of a measurable space are sufficient for constructing the ``complete{''} attraction set corresponding to applying ultrafilters of the family of all subsets of the space of ordinary solutions. We study a compactification of this space that is constructed in the class of Stone ultrafilters (ultrafilters of a measurable space with an algebra of sets) such that the attraction set is realized as a continuous image of the compact set of generalized solutions; we also study the structure of this compact set in terms of free ultrafilters and ordinary solutions that observe the constraints of the problem exactly. We show that, in the case when there are no exact ordinary solutions, this compact set consists of free ultrafilters only; i.e., it is contained in the remainder of the compactifier (an example is given showing that the similar property may be absent for other variants of the extension of the original problem).

Author keywords:

attraction set; extension; ultrafilter

DOI:

10.1134/S0081543811090021

Web of Science ID:

ISI:000297915900002

Соавторы в МНС:

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Другие поля

Поле	Значение
Month	DEC
Publisher	MAIK NAUKA/INTERPERIODICA/SPRINGER
Address	233 SPRING ST, NEW YORK, NY 10013-1578 USA
Language	English
Research-Areas	Mathematics
Web-of-Science-Categories	Mathematics, Applied; Mathematics
Author-Email	chentsov@imm.uran.ru
Funding-Acknowledgement	Russian Foundation for Basic Research {[}09-01-00436, 10-01-96020, 10-01-00356]; Presidium of the Russian Academy of Sciences {[}09-P-1-1007, 09-P-1-1014]
Funding-Text	This work was supported by the Russian Foundation for Basic Research (project nos. 09-01-00436, 10-01-96020, and 10-01-00356) and by the Program for Fundamental Research of the Presidium of the Russian Academy of Sciences ``Mathematical Theory of Control{''} (project nos. 09-P-1-1007 and 09-P-1-1014).
Number-of-Cited-References	24
Usage-Count-Since-2013	5
Journal-ISO	Proc. Steklov Inst. Math.
Doc-Delivery-Number	859ZY