Markov Transition Kernels

Question

Could someone explain how the integral at the bottom of the page is defined? I am generally familiar with measure theory but it may be that I am unfamiliar with the notation. A and x are fixed so are you integrating R(y,A) by the measure defined on $\mathcal{Y}$ by Q(x,-)

Answer 1

$x$ and $A$ are fixed so $ y \to R(y,A)$ is a measurable function; you integrate this measurable function w.r.t. the measure $\mu$ defined by $\mu (E)=Q(x,E)$.