In "Ergodic theory with a view towards number theory" we are asked to show Rohlins lemma holds for aperiodic atomless invertible measure preserving systems.
Not only I can't find a proof, I even don't understand why the following is not a counterexample.
I will build a aperiodic atomless invertible measure preserving system satisfying that there is no nonempty $E$ with $E,T(E)$ disjoint which will cause a contradiction.
First I will give a failed counter-example that will fail being atomless-
A copy of $Z$ with the shift map, the only measureable sets being everything or nothing. Call this the trivial sigma algebra.
To upgrade this to being atomless we consider $[0,1]\times Z$, the sigma algebra is the product sigma algebra of the standard Lebesgue and the trivial sigma algebra. $T$ works by shifting the $Z$ part.
This clearly preserves measure, is atomless since we just think of this as $[0,1]$, is aperiodic, and clearly $E,T(E)$ can't be disjoint.
What am I missing?