Weak measure expansiveness for partially hyperbolic diffeomorphisms

Let M be a closed connected smooth Riemannian manifold with dimM ≥ 2 and let f: M → M be a diffeomorphism. In the paper, we show that C1 generically, if a diffeomorphism f does not present a homoclinic tangency then it is weak Lebesgue measure expansive and, as an example, we find a partially hyperbolic diffeomorphism which is not weak measure expansive. Moreover, for a surface, if a diffeomorphism f has a homoclinic tangency then there is a diffeomorphism g C1 close to f such that g is not weak measure expansive.