Ok, I've already posted this a minute ago, but my text deleted itself while I was editing it :-( So next try:
Can you help me to understand the notation my professor uses to describe Markov processes? Let $(S,\Sigma)$ be a measurable space. We denote the Markov property as $\mathbb{P}(X_{n+1}=x_{n+1}|X_0=x_0,\dots,X_n=x_n)=P(x_n,x_{n+1}),$ so apparently $P$ denotes the transition matrix, here only dependent on $x_n$ and $x_{n+1}$. But then we spoke about the Chapman-Kolmogorov relation, which we denoted by $P_{s+t}(x,B)=\int P_s (x,dy)P_t (y,B),$ where $x\in S$ and $B\in \Sigma$. Have you seen this notation and can you explain to me what it means? (Unfortunately it was not explained in the lecture).
Thank you very much!