Poincaré's Recurrence Theorem counterexample?

Question

I would like to find a counterexample of this Theorem:
"Let $T:X \to X$ be a measure-preserving transformation on a probability space $(X,B,\mu)$, and let $E \subseteq X$ be a measurable set. Then almost every point $x \in E$ return to $E$ infinitely often. That is, there exists a measurable set $F \subseteq E$ with $\mu(F)=\mu(E)$ with the property that for every $x \in F$ there exist integers $0<n_1<n_2<\dots<$ with $T^{n_i}x \in E$ for all $i \geq 1$." I'm aware that Poincaré recurrence is a consequence of the measure space being of finite measure. So we can consider the map $T: \mathbb{R} \to \mathbb{R}$, $T(x)=x+1$. It is known that Lebesgue measure $m$ on $\mathbb{R}$ is invariant by translation. So if we take a bounded set $E \subseteq \mathbb{R}$, for any $x \in E$ the set $\{n \geq 1 | T^{n}x \in E\}$ is finite. (Is this true? How can I formalize it? Is the boundedness of $E$ enough?). Then, if $\{n \geq 1 | T^{n}x \in E\}$ is finite, I can't find a subset $F$ like in the Theorem.

Answer 1

The assertion $T^nx\in E$ is equivalent to $x+n\in E$ and for a bounded set $E$, $E$ and $E-n:=\{e-n,e\in E\}$ are disjoint for $n$ large enough.