Quasi-invariant measures on the orthogonal group over the Hilbert space
In this paper, three subjects in terms of translational-quasi-invariance are discussed for a probability measure \(\mu\) defined on the Borel field \({\mathcal B}\) derived from the strong topology on O(H). O(H) is the orthogonal group over the real Hilbert space with \(\dim (H)=\infty.\) Let G be a subgroup of O(H). First it is shown that ''there does not exist any \(\sigma\) -finite G-quasi-invariant measures'' on \({\mathcal B}\) as far as G acts transitively on the unit sphere of H.'' Next we choose a complete orthonormal system \(e_ 1,...,e_ n,..\). of H, and take G as a following set \(O_ o(H)=\cup^{\infty}_{n=1}O(n),\) where O(n) is a set of consisting of \(T\in O(H)\) which leaves \(e_ p\) invariant for all \(p>n\). Then \(''O_ o(H)\)-quasi-invariant probability measures certainly exist and the proof is carried out with the Schmidt's orthogonalization method''. This is the second subject. The last subject is an ergodic decomposition of \(O_ o(H)\)-quasi-invariant probability measures.
scientific article; zbMATH DE number 3869663