Let $G$ be a locally compact Hausdorff group, $H$ its closed subgroup. To avoid pathologies, we assume the underlying topological space of $G$ has a countable base. Let $\mu$ be a Haar measure on $G$.
Is the following asserion true?
If $\mu(H) \gt 0$, then $H$ is open.
It seems to be false, but I was unable to find a counter-example. For exanple, if $G$ is a Lie group, the assertion seems to be true.