Pre-Image Sigma-Algebra of Generated Sigma-Algebra

Theorem
Let $f: X \to Y$ be a mapping.

Let $\mathcal G \subseteq \mathcal P \left({Y}\right)$ be a collection of subsets of $Y$.

Then the following equality of $\sigma$-algebras on $X$ holds:


 * $f^{-1} \left({\sigma \left({\mathcal G}\right)}\right) = \sigma \left({f^{-1} \left({\mathcal G}\right)}\right)$

where


 * $\sigma$ denotes a generated $\sigma$-algebra
 * $f^{-1} \left({\sigma \left({\mathcal G}\right)}\right)$ denotes the pre-image $\sigma$-algebra
 * $f^{-1} \left({\mathcal G}\right)$ is the preimage of $\mathcal G$ under $f$