Definition:Pullback Finite Partition
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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
Sources
- 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) $4.1$: Partitions and Subalgebras