It is well-known that if a Harris chain $\Phi$ has an invariant probability $\pi$, then we have law of large number, i.e. $$ \lim_{n\rightarrow\infty}\frac1n \sum_{t=1}^n f(\Phi_t)={\rm E}_{\pi}[f(\Phi_0)], ~ a.s.~ {\rm P} $$ for each $f\in L_1$, and ${\rm P}$ is the true probability.
My question: could we obtain a limit in the sense of ${\rm E}_{{\color{red} {\rm P}}}[f(\Phi_0)]$?
Thank you very much!