Measurability with respect to the intersection of two sets does not imply measurability with respect to each set

Question

By counterexample.

Let $\mathcal{F}_{1}, \mathcal{F}_{2}$ be sigma algebras, and $A_{1} \in \mathcal{F}_{1}, A_{2} \in \mathcal{F}_{2}$. Define the indicator functions $\mathbb{1}_{A_{1}}$ and $\mathbb{1}_{A_{2}}$. Then, the product $\mathbb{1}_{A_{1}} \mathbb{1}_{A_{2}} = \mathbb{1}_{A_{1} \cap A_{2}}$ is measurable with respect to $A_{1}\cap A_{2}$ but may not be measurable with respect to either $A_{1}$ or $A_{2}$.

Is this correct?