State and prove the monotone convergence theorem.

Let $\left(E_{1}, \mathcal{E}_{1}, \mu_{1}\right)$ and $\left(E_{2}, \mathcal{E}_{2}, \mu_{2}\right)$ be finite measure spaces. Define the product $\sigma$-algebra $\mathcal{E}=\mathcal{E}_{1} \otimes \mathcal{E}_{2}$ on $E_{1} \times E_{2}$.

Define the product measure $\mu=\mu_{1} \otimes \mu_{2}$ on $\mathcal{E}$, and show carefully that $\mu$ is countably additive.

[You may use without proof any standard facts concerning measurability provided these are clearly stated.]