(i) Define the notions of a $\pi$-system and a $d$-system. State and prove Dynkin's lemma.

(ii) Let $\left(E_{1}, \mathcal{E}_{1}, \mu_{1}\right)$ and $\left(E_{2}, \mathcal{E}_{2}, \mu_{2}\right)$ denote two finite measure spaces. Define the $\sigma$ algebra $\mathcal{E}_{1} \otimes \mathcal{E}_{2}$ and the product measure $\mu_{1} \otimes \mu_{2}$. [You do not need to verify that such a measure exists.] State (without proof) Fubini's Theorem.

(iii) Let $(E, \mathcal{E}, \mu)$ be a measure space, and let $f$ be a non-negative Borel-measurable function. Let $G$ be the subset of $E \times \mathbb{R}$ defined by

$$G=\{(x, y) \in E \times \mathbb{R}: 0 \leqslant y \leqslant f(x)\}$$

Show that $G \in \mathcal{E} \otimes \mathcal{B}(\mathbb{R})$, where $\mathcal{B}(\mathbb{R})$ denotes the Borel $\sigma$-algebra on $\mathbb{R}$. Show further that

$$\int f d \mu=(\mu \otimes \lambda)(G)$$

where $\lambda$ is Lebesgue measure.