On the question of representation of ultrafilters in a product of measurable spaces

We study representations of ultrafilters of widely understood measurable spaces that are realized by means of generalized Cartesian products. The structure of the arising ultrafilter space is established; in more traditional measurable spaces, this structure reduces to the realization of the Stone compact space in the form of a Tikhonov product. The methods developed can be applied to the construction of extensions of abstract attainability problems with constraints of asymptotic nature; in such extensions, ultrafilters can be used as generalized elements, which admits a conceptual analogy with the Stone–Cech compactification. The proposed implementation includes the possibility of using measurable spaces, for which the set of free ultrafilters can be described completely. This results in an exhaustive representation of the corresponding Stone compact space for measurable spaces with algebras of sets. The present issue is devoted to I. I. Eremin's jubilee; the author had many discussion with him on very different topics related to mathematical investigations, and the discussions inevitably led to a deeper understanding of their essence. The author appreciates the possibility of such communication and is grateful to Eremin, who contributed significantly to the development of the mathematical science and education in the Urals.