Quasi-Invariance of Measures on an Infinite Dimensional Vector Space and the Continuity of the Characteristic Functions

Since the Bochner theorem was extended to the infinite dimensional case by Minlos [1] and Sazonov [2], the continuity of a characteristic function has been discussed mainly in connection with the carrier of the corresponding measure. However, the study of the relation between the continuity of a characteristic function and the quasi-invariance of the corresponding measure has been rather neglected. In this paper we shall discuss this problem. Our main results are as follows. Let £ be a vector space, £' be its algebraical dual space, ^ be a finite measure on £', and x be the characteristic function of \i defined on E. Consider the weakest vector topology on E that makes x continuous, and denote it with T^. Let 7J, be the set of all translations on E' under which \i is quasi-invariant. T^ is regarded as a subset of E' by identifying any translation x-»x + a on E' with a. Then we have the following