I need your help to finalise my proof of the following theorem:Let $ (X, B, \mu, T) $ be a measure preserving system, where $(X,B ,\mu)$ is a standard probability space. $ (X, B, \mu, T) $ is a Kronecker factor if and only if it is isomorphic to the ergodic rotation $ R_g(x) = gx $ on a compact abelian group G . I am stuck in specific step:
Proof:
Let $ \{f_i : i \in \mathbb{N}\} $ be the set of eigenfunctions of $ U_T : L^2(\mu) \to L^2(\mu) $ associated with eigenvalues $ \{\lambda_i : i \in \mathbb{N}\} $. Define $ g = (\lambda_1, \lambda_2, \ldots) $. Then the abelian subgroup $ G = \overline{\{g^n : n \in \mathbb{N}\}} \subset \mathbb{S}^\mathbb{N} $ is compact and metrizable. Define the application $ \pi : X \to \mathbb{S}^\mathbb{N}, x \mapsto \left( \frac{f_1}{|f_1|}(x), \frac{f_2}{|f_2|}(x), \ldots \right). $ This is well-defined, injective, and $ \pi(T(x)) = \left( U_T \frac{f_1}{|f_1|}(x), \ldots \right) = \left( \lambda_1 \frac{f_1}{|f_1|}(x), \ldots \right) = g\pi(x) = R_g(\pi(x)) $. Since $ \{f_i : i \in \mathbb{N}\} $ re eigenfunctions, they are measurable by definition,it follows that $\pi$ is measurable.
Let $ \nu = \mu \circ \pi^{-1} |_ G$, using the propreties of the measure $\mu$ and that $\pi$ is measurable, one can conlude the non negativity and the null empty set proprety. Regarding the countable additivity, we will use thet fact that $\pi$ is an injective. Now I just need to prove that $\pi(X)$ is going to be indeed with a coset of G. I was thinking that maybe I can use the fact that $(T^n(x))_{n\in \mathbb{N}}$ is dense in X, but to prove that $\pi((T^n(x))_{n\in \mathbb{N}}$ is also dense in $\pi(X)$, this would be true if $\pi$is continuous, but in this case I just know that $\pi\in L^2$. Any hints or suggestions are most welcome. Thank you !
