Local functionals on C∞( G) and probability

Abstract

A characterization of local functionals on C ∞( G), the space of real continuous functions with compact supports on a locally compact space G, is given. Such functionals were defined by Gel'fand and Vilenkin, as they occur in the analysis of generalized random processes with independent values. The preceding characterization is then used in a representation of the characteristic functionals of the above processes on C ∞( G), and results analogous to those of Gel'fand and Vilenkin on a Schwartz space are obtained. Since C ∞( G) is not nuclear, this study presents new problems and it largely complements the earlier work.