  For a dynamical system, it is known that the existence of a Lyapunov-type
density function, called Lyapunov density or Rantzer's density function,
implies convergence of Lebesgue almost all solutions to an equilibrium. Using
the duality between Frobenius-Perron and Koopmann operators, we generalize this
result from equilibrium to invariant sets, both for continuous- and
discrete-time. Furthermore, some redundant assumptions that exist in the
literature of almost everywhere stability for discrete-time, such as the local
stability of the attractor and the compactness of the state space, has been
removed.
