  In this paper we show the relation between robust transitivity and robust
ergodicity for conservative diffeomorphisms. In dimension 2 robustly transitive
systems are robustly ergodic. For the three dimensional case, we define it
almost robust ergodicity and prove that generically robustly transitive systems
are almost robustly ergodic, if the Lyapunov exponents are nonzero. We also
show in higher dimensions, that under some conditions robust transitivity
implies robust ergodicity.
