\begin{definition}[Definition:Pre-Image Sigma-Algebra/Codomain]
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 codomain of $f$''' is defined as:
:$\Sigma' := \left\{{E' \subseteq X': f^{-1} \left({E'}\right) \in \Sigma}\right\}$
It is a $\sigma$-algebra, as proved on Pre-Image Sigma-Algebra on Codomain is Sigma-Algebra.
\end{definition}