Pre-Image Sigma-Algebra on Domain is Generated by Mapping

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

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

Let $f: X \to X'$ be a mapping.

Then:


 * $\map \sigma f = \map {f^{-1} } {\Sigma'}$

where


 * $\map \sigma f$ denotes the $\sigma$-algebra generated by $f$
 * $\map {f^{-1} } {\Sigma'}$ denotes the pre-image $\sigma$-algebra under $f$

Proof
By Characterization of Sigma-Algebra Generated by Collection of Mappings:


 * $\map \sigma f = \map \sigma {\map {f^{-1} } {\Sigma'} }$

where the latter $\sigma$ denotes a $\sigma$-algebra generated by a collection of subsets.

By Pre-Image Sigma-Algebra on Domain is Sigma-Algebra, $\map {f^{-1} } {\Sigma'}$ is a $\sigma$-algebra.

Hence:
 * $\map \sigma {\map {f^{-1} } {\Sigma'} } = \map {f^{-1} } {\Sigma'}$