Here is a lemma that I know to be true, and can prove in half a page or so, but I'm wondering: can anyone supply a reference so that it can simply be quoted in a paper?

**Lemma** Let $T$ be an ergodic measure-preserving transformation of $(X,\mu)$ and let $n>1$. Then there exists $k$ a factor of $n$ and a set $B$ of measure $1/k$ such that $T^n|_B$ is ergodic. The sets $(T^{-j}B)_{j=0}^{k-1}$ partition $X$.