Definition:Pullback Finite Sigma-Algebra
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Definition
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 the pullback $\sigma$-algebra of $\AA$ by $T^n$ is defined as:
- $ T^{-n} \AA$
that is, it is the preimage $\sigma$-algebra of $\AA$ by $T^n$.
Also see
- Pre-Image Sigma-Algebra on Domain is Sigma-Algebra
- Pullback Finite Sigma-Algebra is Finite Sub-Sigma-Algebra
Sources
- 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) $4.1$: Partitions and Subalgebras