Selecting Universal Partitions in Ergodic Theory

Abstract

Let $\mathscr{P}$ be the set of all $k$-atom measurable partitions of a standard measurable space $(\Omega, \mathscr{F})$, and let $T$ be an isomorphism of $(\Omega, \mathscr{F})$ onto itself. Given $P \in \mathscr{P}$, each probability measure $\mu$ on $\mathscr{F}$ stationary and ergodic with respect to $T$ determines a joint distribution under $\mu$ of the $k$-state stochastic process $(P, T)$. We say that $P$ is universal for a property $S$ (depending on $\mu$) if the distribution of $(P, T)$ satisfies $S$ for all $\mu$. Theorems are given which assure the existence of a universal $P \in \mathscr{P}$.