Invariant boundary distributions for finite graphs

Let $\Gamma$ be the fundamental group of a finite connected graph $\mathcal G$. Let $\mathfrak M$ be an abelian group. A {\it distribution} on the boundary $\partial\Delta$ of the universal covering tree $\Delta$ is an $\mathfrak M$-valued measure defined on clopen sets. If $\mathfrak M$ has no $\chi(\mathcal G)$-torsion then the group of $\Gamma$-invariant distributions on $\partial\Delta$ is isomorphic to $H_1(\mathcal G,\mathfrak M)$.