Alpern Lemma Independent of a Partition for Aperiodic Systems

We prove that for every $(X,\mathcal{B},μ,T)$, an aperiodic, probability preserving system and $\mathcal{P}$ a finite measurable partition of $X$, there exists an Alpern Multitower whose base is independent of $\mathcal{P}$. This removes the ergodicity assumption in the original proof of the indepen