Compactifications of spaces of functions and integration of functionals

Abstract:

For a locally compact space there exists a compactification such that all its points are effectively describable, namely, Alexandroff’s onepoint compactification. The effective construction of compactifications for numerous standard separable metric spaces is already a very nontrivial problem. We propose a method of compactification which enables us to effectively construct compactifications of some spaces of functions (for example, of a ball in ${L_p}( - \infty ,\infty )$). It will be shown that the study of compactifications of spaces of functions is of principal importance in the theory of integration of functionals and in limit theorems for random processes.