Some representations of free ultrafilters

Constructions related to the representation of free $\sigma$-multiplicative ultrafilters of widely interpreted measurable spaces are considered. These constructions are based on the representations connected with the application of open ultrafilters for co-finite and co-countable topologies. Such ultrafilters are preserved (as maximal filters) under the replacement of topologies by algebra and $\sigma$-algebra generated by above-mentioned topologies, respectively. In (general) case of co-countable topology, uniqueness of $\sigma$-multiplicative free ultrafilter composed of nonempty open sets is established. It is demonstrated that the given property is preserved for $\sigma$-algebras containing co-countable topology. Two topologies of the space of bounded finitely additive Borel measures with the property of uniqueness of remainder for sequentially closed set of Dirac measures under the closure construction are stated.