  We formulate a criterion for the existence and uniqueness of an invariant
measure for a Markov process taking values in a Polish phase space. In
addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic
averages of the laws of the process starting from any initial distribution, is
established. The principal assumptions are the existence of a lower bound for
the ergodic averages of the transition probability function and its local
uniform continuity. The latter is called the e-property. The general result is
applied to solutions of some stochastic evolution equations in Hilbert spaces.
As an example, we consider an evolution equation whose solution describes the
Lagrangian observations of the velocity field in the passive tracer model. The
weak-$^*$ mean ergodicity of the corresponding invariant measure is used to
derive the law of large numbers for the trajectory of a tracer.
