Connection between almost everywhere stability of an ODE and advection PDE

A result on the necessary and sufficient conditions for almost everywhere stability of an invariant set in continuous-time dynamical systems is presented. It is shown that the existence of a Lyapunov density is equivalent to the almost everywhere stability of an invariant set. Furthermore, such a density can be obtained as the positive solution of a linear partial differential equation analogous to the positive solution of Lyapunov equation for stable linear systems.