My question is very simple but it confused me when reading this article:
The Metropolitan-Hastings Algorithm and Extensions
See bottom of page $3$, with equation $(2.1.6)$:
$$p(x, y) = a(x, y)q(x, y) + A(x)\delta_x(dy) $$
It is stated that the function $\delta_x(dy)$ is a dirac delta function, but this notation confuses me. Does $\delta_x(dy)$ simply mean the delta function's value estimated at $x=dy$? But $dy$ is an infinitesimal interval? So what's going on here?
For reference, generally this kind of notation is used to denote Lebesgue integration, for example you might write $\int f(x) m(dx)$. This indicates the Lebesgue integral of the function $f$ in the variable $x$ against the measure $m$. The other common alternative is $\int f(x) dm(x)$. One way or the other this notation looks a bit weird because the whole point of it is to indicate the integrand, the integration variable, and the integration measure all in the same expression. Sometimes people prefer to just have a single integration variable which can then be suppressed; this gets written as $\int f dm$. But in complicated multivariate situations it can be too cumbersome to define the $f$ and $m$ that you would need to use this notation.
In this particular context the point is that they're trying to formally write $p(x,y)$ as a transition density even though there is no density, because this measure has a discrete part. This whole expression expands out to
$$P(X_{n+1} \in B \mid X_n=x)=\int_B a(x,y) q(x,y) m(dy) + \int_B A(x) \delta_x(dy)$$
and the last integral is just $A(x)$ if $x \in B$ and zero otherwise. Here $m$ I guess is the Lebesgue measure.