A Radon measure on $G$ being left-invariant on a dense subgroup $H \subset G$ is a Haar measure on $G$.

Question

Let $G$ be a locally compact group, $H$ a dense subgroup, and $μ$ a Radon measure on $G$ such that $μ(hA) = μ(A)$ holds for every measurable set $A ⊂ G$ and every $h ∈ H$. Show that $μ$ is a (left-invariant) Haar measure.

Any help with this is appreciated !