Let $N_1$ and $N_2$ be independent Poisson random measures in (S,$\mathcal{S}$) with intensity measures $\nu_1$ and $\nu_2$, both finite. If $\Pi_1$ and $\Pi_2$ are the supports of $N_1$ and $N_2$, I need to prove that they are disjoint a.s, i.e
$\mathbb{P}(\Pi_1\cap\Pi_2\cap A=\varnothing)=1$
where $A\in\mathcal{S}$. I find myself completely lost with this problem. Could someone guide me to solve it?