I was asked to prove the following:
Let $G$ be a group, equipped with the discrete topology. A Haar measure on $G$ is a measure $\mathcal{P}(G) \rightarrow [0, \infty]$ such that: $$\mu(K) < \infty$$ for all compact sets K and $$\mu(gU) = \mu(U) \; \forall g \in G \; \forall U \subset G$$ Then $\mu = a*z$ where $z$ is the counting measure and $a \in \mathbb{R}_0^+$
The compact sets are all finite sets and therefore this is not too difficult to show if the measure for finite sets is non zero.
But what about the following measures on $\mathbb{R}$ as a group with addition and the discrete topology $$\mu(U) = 0 \; \forall U \subset \mathbb{R}$$ $$\mu(U) = 0 \text{ if U is countable else } \mu(U)=\infty $$ They both seem to be Haar measures to me, but they cant both be a multiple of the counting measure. Am I missing something?