Let $(\Omega,\mathcal{F})$ be a measurable space and let $\mathcal{G}$ be a generator of $\mathcal{F}$, i.e. $\sigma(\mathcal{G}) = \mathcal{F}$. For any subset $A \subset \Omega$ we can form the following $\sigma$-algebra on $\Omega \cap A$: $$\mathcal{F} \cap A = \{B \cap A ~ | ~ B \in \mathcal{F} \} $$
I would like to show that $\mathcal{F} \cap A$ is generated by the family $\mathcal{G} \cap A = \{B \cap A ~ | ~ B \in \mathcal{G} \} $. It is straightforward to see that $$ \sigma(\mathcal{G} \cap A) \subset \mathcal{F}\cap A $$ since $\mathcal{F} \cap A$ is a $\sigma$-algebra containing all sets in the family $\mathcal{G} \cap A$. However, the reverse inclusion is harder to show - and I am not even sure if the statement holds. Any help/hints would be appreciated.
