  We study the invariant measures of typical $C^0$ maps on compact connected
manifolds with or without boundary, and also of typical homeomorphisms. We
prove that the weak$^*$ closure of the set of ergodic measurescoincides with
the weak$^*$ closure of the set of measures supported on periodic orbits and
also coincides withthe set of pseudo-physical measures. Furthermore, we show
that this set has empty interior in the set of invariant measures.
