  The relations between asymptotic stability, the eventual e-property and the
e-property of Markov semigroups, acting on measures defined on general (Polish)
metric spaces, are studied. While usually much attention is paid to asymptotic
stability (and the e-property has been for years verified only to establish
it), it should be noted that the e-property itself is also important as it,
e.g., ensures that numerical errors in simulations are negligible.
  Here, it is shown that any asymptotically stable Markov-Feller semigroup with
an invariant measure such that the interior of its support is non-empty
satisfies the eventual e-property. Moreover, we prove that any Markov-Feller
semigroup, which is strongly stochastically continuous, and which possesses the
eventual e-property, also has the e-property. We also present an example
highlighting that strong stochastic continuity cannot be replaced by its weak
counterpart, unless a state space of a process corresponding to a Markov
semigroup is a compact metric space.
