Let $\,(P_x^t)_{x \in \mathbb{R} , t \geq 0}\,$ be a measurable Markovian family of transition probabilities - that is, a family of Borel probability measures $P_x^t$ on $\mathbb{R}$ such that

1. for all $A \in \mathcal{B}(\mathbb{R})$, the map $(x,t) \mapsto P_x^t(A)$ is Borel-measurable;

2. for all $x \in \mathbb{R}$, $P_x^0=\delta_x$;

3. for all $A \in \mathcal{B}(\mathbb{R})$ and $s,t \geq 0$, $\ P_x^{s+t}(A)=\int_\mathbb{R} P_y^t(A) \, P_x^s(dy)$.

Working over a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t \geq 0}),\mathbb{P})$: suppose we have a progressively measurable real-valued stochastic process $(X_t)_{t \geq 0}$ such that

1. [Markov property] for all $A \in \mathcal{B}(\mathbb{R})$ and $s,t \geq 0$, $\hspace{2mm} \mathbb{E}[\mathbf{1}_A(X_{s+t})|\mathcal{F}_s] \, \overset{\mathbb{P}\textrm{-a.s.}}{=} \, P_{X_s}^t(A)\,$;
2. [stationary] for all $t \geq 0$, $X_{t\ast}\mathbb{P}=X_{0\ast}\mathbb{P}$.

(We emphasise that we have not assumed $(X_t)_{t \geq 0}$ to be right-continuous or even separable.)

Letting $\,\rho:=X_{0\ast}\mathbb{P}\,$, we will say that a set $A \in \mathcal{B}(\mathbb{R})$ is invariant if for each $t \geq 0$, for $\rho$-almost all $x \in \mathbb{R}$, $P_x^t(A)=\mathbf{1}_A(x)$. Let

$\hspace{5mm} \mathcal{G} \ := \{ X_0^{-1}(A) \, : \, \textrm{invariant } A \in \mathcal{B}(\mathbb{R}) \} \; \subset \, \mathcal{F}$.

Fix any bounded measurable $f:\mathbb{R} \to \mathbb{R}$. By the answer to Q1 in Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes, the limit

$\hspace{5mm} L(\omega) \, := \, \lim_{T \to \infty} \frac{1}{T} \int_0^T f(X_t(\omega)) \, dt$

exists for $\mathbb{P}$-almost all $\omega \in \Omega$.

Is it necessarily the case that $\hspace{2mm}\mathbb{E}[f(X_0)|\mathcal{G}] \, \overset{\mathbb{P}\textrm{-a.s.}}{=} \, L \,$?

We can at least start with the more basic question: If $\mathcal{G}$ consists only of null sets and full-measure sets (i.e. $\rho$ is an ergodic measure of $(P_x^t)_{x \in \mathbb{R} , t \geq 0}$), is it necessarily the case that $L$ is almost-everywhere equal to $\int_\mathbb{R} f \, d\rho$?