Construction of limiting process operations using ultrafilters of measurable spaces

Consideration is given to representations of limiting process operations in terms of ultrafilters of measurable spaces and in terms of $(0,1)$-measures generated by these ultrafilters. Properties that are specified by the existence of non-Dirac countably additive $(0,1)$-measures and have the sense of degeneracy of corresponding “limiting” operations are studied. Properties of spaces whose elements are uniform limits of step mappings called tier mappings are analysed. We consider applications of these mappings in constructions connected with the design of correct extensions for abstract problems of attainability that admit natural analogs with problems of construction and study of properties of attainability domains of controlled systems.