Finite measurable partition

A finite measurable partition $\mathcal P=\{P_1,\ldots,P_r\}$ of a probability space consists of measurable cells that are disjoint and cover the space modulo null sets. Cells of measure zero may be discarded. The join $\mathcal P\vee\mathcal Q$ has cells $P\cap Q$ , and $T^{-1}\mathcal P$ has cells $T^{-1}P$ .

Orbit names

The joined partition $\mathcal P^{(n)}=\bigvee_{j=0}^{n-1}T^{-j}\mathcal P$ records the first $n$ cell labels of an orbit. Its cell $\mathcal P^{(n)}(x)$ is the set of points with the same first $n$ labels as $x$ . These finite observations underlie dynamical entropy.