Suppose that $E$ is a metric space, let $\mathcal{B}_E$ denote the set of its Borel subsets and suppose that $\mu$ is a probability measure on $(E,\mathcal{B}_E)$. In addition, suppose that $p:E\times \mathcal{B}_E\to[0,1]$ is a stochastic kernel such that

- For any $B\in\mathcal{B}_E$, $p(\cdot,B):E\to[0,1]$ is a measurable map.
- For any $x\in E$, $p(x,\cdot):\mathcal{B}_E\to[0,1]$ is a probability measure on $(E,\mathcal{B}_E)$.

It can be shown (for example, Lemma 2 on page 11 of this) that for any bounded measurable function $f$

$$\int \left(\int f(y)p(x,dy)\right)\mu(dx)=\int f(x)\nu(dx),\quad\quad(*)$$

where $\nu$ is the probability measure on $(E,\mathcal{B}_E)$ defined by

$$\nu(B):=\int p(x,B)\mu(dx).$$

My question is: If $1\leq q <\infty$ Is it ever true that $(*)$ holds for any $f\in L^q(\mu)$ if $\mu$ is an invariant probability measure of $p$? That is, if for any $B\in \mathcal{B}_E$

$$\mu(B)=\int p(x,B)\mu(dx).$$

The reason I think (hope!) there might be something to the above is that, under some mild conditions on $E$ and $\mu$, one can show that the bounded linear operator on the space of bounded measurable functions defined by $p$ can be extended to $L^q(\mu)$.