Let $S$ be Polish, $\mu\in M_1(S)$ and let $P_tf(x)=\int f(y)\,P_t(x,dy)$ be a continuous Feller semigroup on $C_b(S)$. Then $P_t^*\nu(A)=\P^\nu(X_t\in A)$. If there is a probability measure $\nu$ such that for all $\e > 0$ there is a compact set $K$ such that for all $t > 0$: $\P^\nu(X_0\in K,X_t\in K^c) < \e$, then the family $P_t^*\nu$, $t > 0$, is relatively compact in $M_1(S)$ and there is an invariant probability measure $\mu$ in $M_1(S)$, i.e. $P_t^*\mu=\mu$ for all $t > 0$.
2. Moreover, if there is only one invariant probability measure $\mu\in M_1(S)$, then for all $f\in C_b(S)$ $A_tf$ converges pointwise to $\int f\,d\mu$. Hence $P_t$ is ergodic in $L_1(\mu)$.
1. The set
$$
C\colon=\cl{\convex{P_t^*\nu:t\geq0}}
$$
is a compact and convex subset of $M_1(S)$. By the Markov Kakutani Theorem there is some $\mu\in C$ such that for all $t$: $P_t^*\mu=\mu$.
2. For any $\nu\in M_1(S)$ and any $f\in C_b(S)$ we have
\begin{eqnarray*}
\int f\,dP_s^*A_t^*\nu-\int f\,dA_t^*\nu
&=&\frac1t\int_{s}^{t+s}\int f\,dP_r^*\nu\,dr
-\frac1t\int_0^t\int f\,dP_r^*\nu\,dr\\
&=&\frac1t\int_t^{t+s}\int f\,dP_r^*\nu\,dr
-\int_0^s\int f\,dP_r^*\nu\,dr
\leq\frac{2s}t\norm f~.
\end{eqnarray*}
Thus if $\mu$ is any accumulation point of the filterbasis $\{A_t^*\nu:t > n\}$, then
$\mu$ is invariant. If there is only one invariant probability measure, then $A_t^*\nu$ converges in $M_1(S)$ to $\mu$ as $t$ converges to $\infty$. In particular for $\nu=\d_x$:
$$
\lim_t A_tf(x)
=\lim_t\int A_tf\,d\d_x
=\lim_t\int f\,dA_t^*\d_x
=\int f\,d\mu~.
$$