Let $G$ be a compact connected topological group and let $H$ be a subgroup of $G$. Suppose that $H$ is measurable with respect to the normalised Haar measure $\mu$ on $G$. Do we necessarily have $\mu(H)=0$ or $\mu(H)=1$?
Maybe this is well--known, I ask it just out of curiosity. The question is related to this one: If you provide a measurable subgroup $H$ of $\mathbb R/\mathbb Z$ of measure not 0 or 1, then the characteristic function of $H$ violates the conjecture stated there.