Generator Criterion for Measurability

Let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be measurable spaces and let $f:X\to Y$ be a function.

1. (Pullback $\sigma$-algebra) The family of all subsets $B$ of $Y$ with $f^{-1}(B)\in\mathcal{F}$ is a $\sigma$-algebra on $Y$.

2. (Generator criterion) Let $\mathcal{C}$ be a family of subsets of $Y$ such that $\mathcal{G}$ is the $\sigma$-algebra generated by $\mathcal{C}$. If $f^{-1}(C)\in\mathcal{F}$ for every $C\in\mathcal{C}$, then $f$ is measurable with respect to $\mathcal{F}$ and $\mathcal{G}$.