Are transition probabilities always absolutely continuous w.r.t. invariant measure?

Question

Let $X_n$ be a Markov process with transition kernel $p$. A probability measure $\mu$ is called invariant (or stationary) if $$ \int p(x,A)\,d\mu(x) = \mu(A) $$ for all measurable sets $A$. My question is: does this imply that the measure $p(x,\cdot)$ is absolutely continuous w.r.t. $\mu$ for $\mu$-a.e. $x$? If $mu(A)=0$, then the invariance gives me $p(x,A)=0$ for $\mu$-a.e. $x$. But the exceptional null set here depends on $A$. So, the question is: Is there an exceptional null set $N$ so that for $x\notin N$ we have $\mu(A)=0$ implies $p(x,A)=0$? We may assume that $\mu$ is the unique invariant measure for the process.

Answer 1

The answer is no. For example, consider a simple random on $\mathbb{R}/\mathbb{Z}$ with steps $\pm \alpha$, each possibility having probability $1/2$. The probabilities $p(x,\cdot) = (\delta_{x+\alpha}+\delta_{x-\alpha})/2$ are not absolutely continuous with regard to the unique invariant measure which is Haar measure on $\mathbb{R}/\mathbb{Z}$.