Is there a locally compact group $G$ and a Borel measure $\mu$ on $G$ such that \begin{equation*} H=\{g\in G\mid \mu(gE)=\mu(E) \: \text{for all measurable} \: E\} \end{equation*} is not a closed subgroup of $G$ ?
Theorem 62.G in Halmos' textbook states that if $\mu$ is a Baire measure then $H$ is always closed. So a counter example must involve a group $G$ which isn't first countable.
I don't know any such groups except those of the form $G^{I}$ with $I$ uncountable and $G$ some usual group. With those I tried to construct a measure so that $H$ is the set of elements with finitely many non trivial coordinates, but it wouldn't be Borel (finite on compact sets).
