Various expansive measures for flows

We discuss a characterization of countably expansive flows in measure-theoretical terms as in the discrete case [2]. More precisely, we define the countably expansive flows and prove that a homeomorphism of a compact metric space is countable expansive just when its suspension flow is. Moreover, we exhibit a measure-expansive flow (in the sense of [4]) which is not countably expansive. Next we define the weak expansive measures for flows and prove that a flow of a compact metric space is countable expansive if and only if it is weak measure-expansive (i.e. every orbit-vanishing measure is weak expansive). Furthermore, unlike the measure-expansive ones, the weak measure-expansive flows may exist on closed surfaces. Finally, it is shown that the integrated flow of a C1 vector field on a compact smooth manifold is C1 stably expansive if and only if it is C1 stably weak measure-expansive.