Uniqueness of the solution to the Vlasov-Poisson system with bounded density

In this note, we show uniqueness of weak solutions to the Vlasov-Poisson system on the only condition that the macroscopic density $\rho$ defined by $\rho(t,x) = \int_{\Rd} f(t,x,\xi)d\xi$ is bounded in $\Linf$. Our proof is based on optimal transportation.