Let $(M_n)_{n\geq0}$ denote a Markov chain with continuous state space and $X$ some $\mathbb{N}$-valued random variable. To my knowledge, the transition probabilities of $M$ are usually described by transition kernels.
Question 1: In a paper I read the transition kernels were written as: $P (x; dy)$. Is it correct that the "common" notation for this would be $\kappa (x, dy)$?
Question 2: Moreover, is it legitimate to write \begin{align} P(X = x, M_n \in dy) = P(X=x \vert M_n = y) P(M_n \in dy), \end{align} or is this abuse of notation or simply not correct?
Remark: I am aware that sometimes $M$ is not called a Markov chain anymore as its state space is continuous, but don't let it bother you here.