Let $E$ be a set and $\mathcal{E} \subseteq \mathcal{P}(E)$ be a set system.

(a) Explain what is meant by a $\pi$-system, a $d$-system and a $\sigma$-algebra.

(b) Show that $\mathcal{E}$ is a $\sigma$-algebra if and only if $\mathcal{E}$ is a $\pi$-system and a $d$-system.

(c) Which of the following set systems $\mathcal{E}_{1}, \mathcal{E}_{2}, \mathcal{E}_{3}$ are $\pi$-systems, $d$-systems or $\sigma$-algebras? Justify your answers. ( $\#(A)$ denotes the number of elements in $A$.)

$E_{1}=\{1,2, \ldots, 10\}$ and $\mathcal{E}_{1}=\left\{A \subseteq E_{1}: \#(A)\right.$ is even $\}$,

$E_{2}=\mathbb{N}=\{1,2, \ldots\}$ and $\mathcal{E}_{2}=\left\{A \subseteq E_{2}: \#(A)\right.$ is even or $\left.\#(A)=\infty\right\}$,

$E_{3}=\mathbb{R}$ and $\mathcal{E}_{3}=\{(a, b): a, b \in \mathbb{R}, a<b\} \cup\{\emptyset\} .$

(d) State and prove the theorem on the uniqueness of extension of a measure.

[You may use standard results from the lectures without proof, provided they are clearly stated.]