Transition Kernel as Operator on an Integrable function

Question

I'm reading "Monte Carlo Statistical Methods" by Robert and Casella. On Pg 211, in the chapter "Markov Chains," we have the following statement:

$$Kh(x) = \int h(y) K(x, dy)$$

where $K$ is a transition kernel and $h$ is an integrable function.

I don't understand how to interpret $dy$ in this since it is an argument to the transition kernel. What is the intuition behind this?

Answer 1

So, if $K$ is a Markov kernel, then $K(x,\cdot)$ is measure, while $K(\cdot, A)$ is a measurable function. Hence, $\int h(y) K(x,\textrm{d}y)$ means integrate $h$ with respect to the measure $K(x,\cdot)$. Measurability of $K$ in $x$ then guarantees measurability of the resulting function.