On one example of representing the ultrafilter space for an algebra of sets

Abstract problems on attainability with constraints of asymptotic nature often involve a situation when the class of sequential approximate sequence solutions (which corresponds conceptually to Vargas approach in control theory problems) is insufficient for the reproduction of effects related to the realization of limit states corresponding to the observance of asymptotic constraints. In this situation, it is necessary to use filters or nets in the original space of solutions. In the case of using filters, as easily seen, it is sufficient to take ultrafilters as analogs of Vargas approximate solutions. However, free ultrafilters, which are the most interesting form this point of view variants of ultrafilters, do not admit a constructive description. The situation can be corrected in some cases of using ultrafilters of an algebra of sets, which turns out to be acceptable in some problems of the above type. In this context, classes of measurable spaces with algebras (or, which is practically the same, with semialgebras) of sets are of interest, as they can be used to describe the set of all free ultrafilters. We analyze an example of this kind and discuss some general constructions related to representations of the space of ultrafilters.