# Kernel for a markov process?

Could anyone just explain to me what does it mean by mathematically, $$P_n(x, dy)$$ is the law of $$X_n$$ here in the page $$46$$.

https://statweb.stanford.edu/~cgates/PERSI/papers/iterate.pdf

Thanks for helping.

They full quote is "The kernel $$P_n(x, dy)$$ is the law of $$X_n$$ given that $$X_0 = x$$." This means that the conditional distribution of $$X_n$$ given $$X_0 = x$$ is given by $$\Pr(X_n \in B \mid X_0 = x) = P_n(x, B)$$ for all measurable subsets $$B$$ of the state space. The word "law" is a synonym for "distribution" in this context. The "$$dy$$" notation comes from the fact that for a sufficiently nice measurable function $$g$$ we have $$E[g(X_n) \mid X_0 = x] = \int_{\mathcal{X}} g(y) P_n(x, dy),$$ where $$\mathcal{X}$$ is the state space (I didn't check what notation the link uses for the state space).