\begin{definition}[Definition:Pre-Image Sigma-Algebra/Domain]
Let $X, X'$ be sets.
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:
:$f^{-1} \sqbrk {\Sigma'} := \set {\map {f^{-1} } {E'}: E' \in \Sigma'}$
It is a $\sigma$-algebra, as proved on Pre-Image Sigma-Algebra on Domain is Sigma-Algebra.
\end{definition}