Quasi-Invariant Measures on the Orthogonal Group over the Hilbert Space

Abstract

Let H be a real separable Hilbert space and be the orthogonal group over H. In this paper, we shall discuss left, right or both trans! at ion ally quasi-invariant probability measures on a afield S3 derived from the strong topology on O(H). Invariant (rather than quasi-invariant) measures have been considered by several authors. For example in [3], [7] and [4] such measures were constructed as suitable limits of Haar measures on O(n) by methods of Schmidt's orthogonalization or of Cayley transformatioiie And in [6] some approach based on Gaussian measures on infinite-dimensional linear spaces was attempted. However these measures are defined on larger spaces rather than and invariant under a sense that O(H) acts on these spaces. This is reasonable, because it is impossible to construct measures on which are invariant under all translations cf elements of G, if G is a suitably large subgroup of O(H). For example, let el9 • • • , en, • • be a c. o. n. s. in H, and for each n consider a subgroup consisting of T^O(H) which leaves ep invariant for all p^>n. We may identify this subgroup with O(ri). Put O0(H) — U ~=1OOz). Then OQ(fT)-invariant finite measure does not exist on O(fT). (See, [6]). However replacing invariance with quasi-invariance, the above situation becomes somewhat different. One but main purpose of this paper is to indicate this point. We will show that there does not exist any a-finite G-quasi-invariant measure on S3, as far as G acts transitively on the unit sphere S of H. While O0(H)-quasi-invariant probability measures certainly exist.