Generator for Product Sigma-Algebra

Theorem
Let $\left({X, \Sigma_1}\right)$ and $\left({Y, \Sigma_2}\right)$ be measurable spaces.

Let $\mathcal G_1$ and $\mathcal G_2$ be generators for $\Sigma_1$ and $\Sigma_2$, respectively.

Then $\mathcal G_1 \times \mathcal G_2$ is a generator for the product $\sigma$-algebra $\Sigma_1 \otimes \Sigma_2$.