# Rohlin lemma for aperiodic atomless invertible measure preserving systems

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?

• You did not specify the measure. After that you need to verify that you have a Lebesgue space. – John B Mar 26 at 0:20