  We prove that a homeomorphism of a compact metric space has an expansive
measure \cite{ms} if and only if it has many ones with invariant support. We
also study homeomorphisms for which the expansive measures are dense in the
space of Borel probability measures. It is proved that these homeomorphisms
exhibit a dense set of Borel probability measures which are expansive with full
support. Therefore, their sets of heteroclinic points has no interior and the
spaces supporting them have no isolated points.
