\[ \newcommand{\C}{\mathbb{C}} \newcommand{\haar}{\mathsf{m}} \newcommand{\P}{\mathcal{P}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\d}{\mathsf{d}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \newcommand{\B}{\mathscr{B}} \]

1. Let $G$ be a finite group. Define $\mu : \P(G) \to [0,\infty]$ by \[ \mu(E) = \dfrac{|E|}{|G|} \] for all $E \subset G$ where $|\,\cdot\,|$ denotes cardinality.
   1. Check that $\mu$ is a measure.
   2. If $H$ is a subgroup of $G$ what is $\mu(H)$?
   3. Prove that $\mu$ is invariant in the sense that $\mu( g^{-1} E) = \mu(E)$ where \[ g^{-1} E = \{ h \in G : gh \in E \} \] for all $E \subset G$ and all $g \in G$.
   4. Are there any other invariant measures on $G$?

2. Define $\nu$ on $\mathcal{P}(\N)$ by \[ \nu(E) = \limsup_{N \to \infty} \dfrac{| E \cap \{1,\dots,N\}|}{N} \] for all $E \subset \N$.
   1. Calculate $\nu(\N)$, $\nu( 2\N )$ and $\nu(\{ n^2 : n \in \N \})$.
   2. Is $\nu$ a measure?
   3. Describe explicitly what measures on $\N$ are like.