Is it true that every subgroup of finite non-zero Haar measure of an abelian locally compact group should be open and compact? This is obviously true for the case of discrete abelian groups. Thanks.
Let $G$ be an LCA group, $\mu$ be a Haar measure on $G$ and $H$ be a (closed) subgroup of $G$ with $0<\mu(H)<\infty$. First, by restricting $\mu$ to $H$, we obtain a finite Haar measure on $H$. This means that $H$ is compact. Further, by projecting $\mu$ to $G/H$ via the quotient morphism $G\to G/H$, we obtain a Haar measure $\nu$ on $G/H$, which is positive on singletons. This means that $G/H$ is discrete, hence $H$ is open in $G$.