Consider a discrete time Markov chain $(X_t)$ on a finite state space $\mathcal{S}$, with transition matrix $P$. Assume that the chain admits a stationary distribution $\pi$, which I will identify with its probability mass function $(\pi(i))_{i\in \mathcal{S}}$. In this case I know that, given any function $f:\mathcal{S}\to\mathbb{R}$, the following ergodic theorem holds $$\mathbb{P}\left(\lim_{n\to\infty}\frac{1}{n}\sum_{t=0}^n f(X_t)=\sum_{i\in\mathcal{S}}f(i)\pi(i)\right)=1$$ I would be interested to know whether, and under which hypotheses, the result generalize to the case when:

- $\mathcal{S}$ is a compact metric space (in particular $\mathcal{S}=[0,1]$). Of course, here the chain is described by a transition kernel $P:\mathcal{S}\to\Delta(\mathcal{S})$ (with the notation $\Delta(\mathcal{S})$ I mean the space of probability distributions on $\mathcal{S}$) and its stationary distribution, if it exists, is $\pi\in\Delta(\mathcal{S})$ such that $\pi=\int_{\mathcal{S}}P(s)d\pi(s)$
- Instead of considering the time average of a function $f:\mathcal{S}\to\mathbb{R}$, we consider the time average of the (independent) realizations of a family of random variables whose law depends on the state of the chain, namely a Markov kernel $\Phi:\mathcal{S}\to\Delta(\mathbb{R})$.

It makes sense to ask that, for all $s\in\mathcal{S}$ it holds $\int_{\mathbb{R}}xd\Phi(s)(x)<\infty$, so that each variable in the family has finite expectation. Denote by $\phi(s)$ a realization of $\Phi(s)$.

My **question** is: (under what hypotheses) does it still hold that

$$\mathbb{P}\left(\lim_{n\to\infty}\frac{1}{n}\sum_{t=0}^n \phi(X_t)=\int_\mathcal{S}\left(\int_{\mathbb{R}}xd\Phi(s)(x)\right)d\pi(s)\right)=1$$

**Observation**: what I am able to observe if a version of the ergodic theorem holds for continuous state spaces, then, trivially, defining $F:\mathcal{S}\to\mathbb{R}$ such that $F(s)=\int_{\mathbb{R}}xd\Phi(s)(x)$, my statement of interest is true if
$$\lim_{n\to\infty}\frac{1}{n}\sum_{t=0}^n \phi(X_t)=\lim_{n\to\infty}\frac{1}{n}\sum_{t=0}^n F(X_t)$$
This seems to point to a relation between the kernels $P$ and $\Phi$. But I don't really know which.

Any help or reference would be greatly appreciated. I beg your pardon in advance in case my notation is not precise, and I am willing to clarify it.