Converse theorem for almost everywhere stability using Lyapunov measure

In our recent paper [1][2], Lyapunov measure is introduced as a new tool for verifying almost everywhere stability of an invariant set in dynamical systems and continuous mapping. In this paper we show that the existence of Lyapunov measure is both necessary and sufficient for almost everywhere stability. The necessary and sufficient condition for almost everywhere stability using Lyapunov measure is analogous to necessary and sufficient condition for asymptotic stability in linear system. In particular the finite dimensional matrix Lyapunov equation for verifying stability in linear systems is replaced by infinite dimensional linear equation for verifying almost everywhere stability of an invariant set in nonlinear systems.