In order to apply a theorem in the lecture my professor used the following lemma that states what follows: One can assume without loss of generality that the system obtained in Frustenberg's Correspondence Principle is ergodic and has continuous eigenfunctions.
Question He didn't give to us the proof of this Lemma, and I don't see why this is true! I was wondering how to prove it. Does somebody see how to prove it and has some hints?
I recall the Furstenberg's Correspondence Principle: For any $ A \subseteq \mathbb{N} $ there exists a compact metric space $X$, a Borel probability measure $\mu$ on $X$, a continuous measure preserving transformation $T:X \to X$, a point $x \in X $ that is generic for $\mu$ along some sequence $(I_k)_{k\in \mathbb{N}}$ of intervals whose length dents to infinity, and a clopen set $E \subseteq X$ such that $\overline{d}(A) = \mu (E) $ and $A= \{ n \in \mathbb{N} : T^n x \in E \} $, where $\overline{d}(A)$ is the upper density of $A$.