Show that either $\mu(S) = 0$ or $\mu(G \setminus S) = 0$

Question

Let $G$ be a locally compact group, $H \leq G$ be a dense subgroup and $\mu$ be a Haar measure.

Let $S \subseteq G$ be a measurable subset such that for each $h \in H$ the sets $hS \cap (G \setminus S)$ and $S \cap (G \setminus hS)$ are both null-sets.

Then, show that either $\mu(S) = 0$ or $ \mu(G \setminus S) = 0$.

Any help with this question!