S. Alpern has proved that an invertible antiperiodic
measurable measure preserving transformation of a Lebesgue
probability space can be represented by
$k$ towers of heights
$n_1, \ldots, n_k$, with prescribed measures, provided
that the heights have greatest common divisor 1. In this
paper we give a simple proof of Alpern's theorem. It is
elementary in the sense that it involves no limits and uses
Kakutani's easy proof of Rokhlin's Lemma.