To question about representation of Stone compactums

We consider the questions connected with the representation of ultrafilters of measurable spaces and finitely additive $(0,1)$-measures for consequent application in extension constructions of abstract attainability problems and extremal problems. Properties connected with the application of (generalized) Cartesian products and their subspaces, and the property having the sense of the identification of ultrafilters and finitely additive $(0,1)$-measures and realized in the form of homeomorphism of natural topologies are investigated.