On the Law of Large Numbers for Markov Processes

The main result of this paper is the derivation of the law of large numbers for Markov processes. More exactly, let $\lambda$ be a sub-invariant measure for a measurable Markov process $(x_t,\mathcal{M}_t,P_x)$ and let $H$ be the Hilbert space of functions $f$ which satisfy the condition $\int{{|f|}^2 d\lambda}<\infty$. Then there exists the limit (in the norm of $H$)
$$\mathop{\lim}\limits_{T\to\infty}\frac{1}{T}\int_0^T{M_x f(x_t )\,dt=g(x)}$$
and we have for any $\varepsilon>0$
$$\mathop{\lim}\limits_{T\to\infty}P_\lambda\left\{{\left|{\frac{1}{T}\int_0^T{f(x_t)\,dt-g(x_0)}}\right|>\varepsilon}\right\}0,$$
where
$$P_\lambda\{\cdot\}=\int{P_x\{\cdot\}\lambda(dx)}.$$