  Given a measure-preserving transformation $T$ of a probability space $(X,
\mathcal B, \mu)$ and a finite measurable partition $P$ of $X$, we show how to
construct an Alpern tower of any height whose base is independent of the
partition $P$. That is, given $N \in \N$, there exists a Rohlin tower of height
$N$, with base $B$ and error set $E$, so that $B$ is independent of $P$, and
$T(E) \subset B$.
