# Seeking Help with a Proof Step in Measure-Preserving Systems and Kronecker Factors

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 !