Definition:Pre-Image Sigma-Algebra/Domain

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

Let $\Sigma'$ be a $\sigma$-algebra on $X'$.

Then the pre-image $\sigma$-algebra (of $\Sigma'$) on the domain of $f$ is defined as:


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

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

Also known as
As usual, one may also write pre-image sigma-algebra.

Sometimes, this is plainly called the pre-image $\sigma$-algebra, but this leads to confusion with the pre-image $\sigma$-algebra on the codomain.