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 $\rho (t,x)={\int }_{\text{Rd}}f(t,x,\xi )d\xi$ is bounded in $\text{Linf}$. Our proof is based on optimal transportation.