In the MC and MCMC literature one commonly finds statements of the following form (see e.g. Roberts & Rosenthal, 2004):
$$ \int_{x \in \mathcal X} \pi(dx)P(x, dy) = \pi(dy). $$
What is the motivation for this type of notation?
In particular I find the following points interesting:
- Why does the measure $\pi$ appear to the left of the kernel $P(x, dy)$? That is, why not follow the convention that the integrating measure appears to the right, viz. $\int f(x) \mu(dx)$?
- What's the precise meaning of the statement? In the text referenced, the statement is proceeded by the clause that it holds for all $x,y\in\mathcal X$. I can't make sense of this. The statement is not a function of $x$ as I read it, and I don't see how the $y$ in $dy$ matters. I'm not sure I even understand what $dy$ even means in this setting.
My interpretation of the statement is the following:
$$ \int_\mathcal{X}P(x, A)\pi(dx) = \pi(A), $$
for every $A$ in the relevant $\sigma-$field on $\mathcal X$.