I have recently taken a course on topological groups and their Haar measure and I should first mention I am still a beginner so even though I thought about this for a while, I was hoping someone here could please help me:
Let $ Q $ have the relative topology induced from $ R $. Then we are to show that the group Q is not locally compact, and also that there is no nonzero translation-invariant Borel measure on $ Q $ that is finite on compact sets.
OK here is what I have got: proving that Q is not locally compact as a topological group with the subspace topology was alright by using the Baire category theorem, that is fine. However, showing that there is no non-zero translation-invariant Borel measure which is finite on compact sets I could not do and to be honest I would truly appreciate someone showing me how to do this part. Thanks all.
Suppose $\mu$ is a translation-invariant Borel measure. By translation-invariance, $\mu(\{x\})=C$ is the same for all $x\in\mathbb{Q}$. Since every subset of $\mathbb{Q}$ is countable, the countable additivity of $\mu$ implies $\mu(A)=|A|C$ for all $A\subseteq\mathbb{Q}$ (where $|A|$ is the cardinality of $A$, and we take the usual measure-theoretic convention that $\infty\cdot 0=0$). So if $C=0$, then $\mu=0$. But if $C>0$, then taking $A$ to be any infinite compact subset of $\mathbb{Q}$, we find that $\mu$ is not finite on compact sets.