Filters and ultrafilters in the constructions of attraction sets

The abstract attainability problem with constraints of asymptotic character is considered; for this problem, the nonsequential (generally speaking) attraction set obtained by the comparison to the solution an attraction element is constructed. Self solutions are defined as nets, filters, or ultrafilters of the space of usual solutions (each of the above-mentioned classes is sufficient for constructing of attraction set). Main attention are given to questions of constructing of attraction sets in the class of measurable spaces interpreted very broad (spaces with the families closed with respect to intersections, measurable spaces with algebras of sets and so forth). The construction arising under consideration of ultrafilters of lattices of sets is used as instrument of investigation.