Certain problems of asymptotic analysis and extensions. I

Asymptotic attainability interpreted as the approximate realization of desirable states under weak slackening of constraints is studied. Constructs ensuring the realization of asymptotically attainable states in terms of standard constraints on the choice of generalized elements and interpreted as finitely additive measures on a suitable measurable space are designed. Special attention is paid to “Boolean-valued” realizations of generalized problems (extensions) in the class of (0,1)-measures or ultrafilters of measurable spaces and are interpreted as direct extension constructs (in the sense of transitions from exact solutions to approximate solutions).