Definition:Pre-Image Sigma-Algebra/Domain

Definition
Let $X, X'$ be sets, and let $f: X \to X'$ be a mapping.

Let $\mathcal{A}'$ be a $\sigma$-algebra on $X'$.

Then the pre-image $\sigma$-algebra (or preimage $\sigma$-algebra) (of $\mathcal{A}'$ by $f$) is defined as:


 * $f^{-1} \left({\mathcal{A}'}\right) := \left\{{f^{-1} \left({A'}\right): A' \in \mathcal{A}'}\right\}$

It is a $\sigma$-algebra, as proved on Pre-Image Sigma-Algebra is Sigma-Algebra.