Definition:Pullback Finite Partition

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\xi$ be a partition of $\Omega$.

Let $T: X \to X$ be a measurable mapping.

Let $n \in \N$.

Then the pullback partition of $\xi$ by $T^n$ is defined as:
 * $ T^{-n} \xi := \set {T^{-n} \sqbrk {A_1}, \ldots, T^{-n} \sqbrk {A_k} } \setminus \set \O$

where:
 * $\xi = \set {A_1, \ldots, A_k}$
 * $T^{-n} \sqbrk A$ denotes the preimage of the set $A$ under the power $T^n$.

Also see

 * Pullback Finite Partition is Finite Partition