Some New Examples of Quasi-Invariant Measures on a Hilbert Space

Abstract

In this paper, we shall consider the translationally quasi-invariant measures on infinite-dimensional linear topological spaces, especially on a rigged Hilbert space Ec:Hc:E*. Let \.i be a measure on E*. We say that a measure \i is quasi-invariant, if n(A) = Q implies ^(A + e) = Q for all eeE*. After this definition, in the finite dimensional case we can characterize \JL as the Lebesgue measure modulo equivalence of absolute continuity. But in the infinite-dimensional case this definition is unsuccessful, because there does not exist such measure except trivial one. On the other hand if we consider only those translations which are defined by the elements of E or H, then there exist continuously many quasi-invariant measures which are singular with respect to each other. However the only explicit known example was the measure of Gaussian type up to the present time. But here we shall give two examples of translationally quasi-invariant ergodic measures which are essentially different from Gaussian ones. This is the purpose of the present paper. The author thanks to Professor H. Yoshizawa for the many valuable comments.