## The Annals of Probability

### 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 $P \in \mathscr{P}$.

#### Article information

Source
Ann. Probab., Volume 9, Number 4 (1981), 705-709.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994379

Digital Object Identifier
doi:10.1214/aop/1176994379

Mathematical Reviews number (MathSciNet)
MR624699

Zentralblatt MATH identifier
0464.60036

JSTOR

Subjects
Primary: 28A65
Secondary: 60G10: Stationary processes 94A15: Information theory, general [See also 62B10, 81P94]

#### Citation

Kieffer, John C.; Rahe, Maurice. Selecting Universal Partitions in Ergodic Theory. Ann. Probab. 9 (1981), no. 4, 705--709. doi:10.1214/aop/1176994379. https://projecteuclid.org/euclid.aop/1176994379