Definition:Pullback Finite Sigma-Algebra

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