Week 2 Tutorial

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

Define $ν$ on $P (N)$ by $ν (E) = \underset{N \to \infty}{lim sup} \frac{| E \cap {1, \dots, N} |}{N}$ for all $E \subset N$ .
1. Calculate $ν (N)$ , $ν (2 N)$ and $ν ({n^{2} : n \in N})$ .
2. Is $ν$ a measure?
3. Describe explicitly what measures on $N$ are like.