Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4590687 | Journal of Functional Analysis | 2012 | 11 Pages |
We will demonstrate that if M is an uncountable compact metric space, then there is an action of the Polish group of all continuous functions from M to U(1) on a separable probability algebra which preserves the measure and yet does not admit a point realization in the sense of Mackey. This is achieved by exhibiting a strong form of ergodicity of the Boolean action known as whirliness. This is in contrast with Mackeyʼs point realization theorem, which asserts that any measure preserving Boolean action of a locally compact second countable group on a separable probability algebra can be realized as an action on the points of the associated probability space. In the course of proving the main theorem, we will prove a result concerning the infinite-dimensional Gaussian measure space (RN,γ∞) which is in contrast with the Cameron–Martin Theorem.