In this paper, the authors claim that
Theorem 1. A random point process $\Pi$ on a regular measure space is a Poisson process if and only if $N_\Pi$ defined by $N_\Pi(A) = \#\{\Pi \cap A\}$ is a completely random measure. If this is true, the mean measure is given by $\mu(A) = \mathbb{E}(\Pi(N(A))$.
They mention that this result can be proven using the Lévy-Khinchin representation, but I could not find a formal proof in any references.