Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems

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).