Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. For $\mathcal{G} \subseteq \mathcal{F}$, what is meant by saying that $\mathcal{G}$ is a $\pi$-system? State the 'uniqueness of extension' theorem for measures on $\sigma(\mathcal{G})$ having given values on $\mathcal{G}$.

For $\mathcal{G}, \mathcal{H} \subseteq \mathcal{F}$, we call $\mathcal{G}, \mathcal{H}$ independent if

$$\mathbb{P}(G \cap H)=\mathbb{P}(G) \mathbb{P}(H) \quad \text { for all } \quad G \in \mathcal{G}, H \in \mathcal{H}$$

If $\mathcal{G}$ and $\mathcal{H}$ are independent $\pi$-systems, show that $\sigma(\mathcal{G})$ and $\sigma(\mathcal{H})$ are independent.

Let $Y_{1}, Y_{2}, \ldots, Y_{m}, Z_{1}, Z_{2}, \ldots, Z_{n}$ be independent random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. Show that the $\sigma$-fields $\sigma(Y)=\sigma\left(Y_{1}, Y_{2}, \ldots, Y_{m}\right)$ and $\sigma(Z)=\sigma\left(Z_{1}, Z_{2}, \ldots, Z_{n}\right)$ are independent.