# Pullback Commutes with Generating Partition

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## Theorem

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

Let $\AA \subseteq \Sigma$ be a finite sub-$\sigma$-algebra.

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

Let $n \in \N$.

Then:

- $T^{-n} \map \xi \AA = \map \xi {T^{-n} \AA}$

where:

- $\map \xi \cdot$ denotes the generated finite partition
- $\map {T^{-n}} \AA$ denotes the pullback partition of $\AA$ by $T^n$
- $T^{-n} \map \xi \AA$ denotes the pullback partition of $\map \xi \AA$ by $T^n$

## Proof

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## Sources

- 2013: Peter Walters:
*An Introduction to Ergodic Theory*(4th ed.) $4.1$: Partitions and Subalgebras