  In this paper we investigate the relation between measure expansiveness and
hyperbolicity. We prove that non atomic invariant ergodic measures with all of
its Lyapunov exponents positive is positively measure-expansive. We also prove
that local diffeomorphisms robustly positively measure-expansive is expanding.
Finally, we prove that if a $C^1$ volume preserving diffeomorphism that. can
not be accumulated by positively measure expansive diffeomorphis have a
dominated sppliting.
