There is a formulation for Markov chains that I don't understand. Consider a Markov Chain $\textbf{X}=\{X_n:n=0,1,...\}$, with arbitrary state space $S$ with countably generated $\sigma$-field $\mathcal{B}$.
I know there is this post explaining what a countably generated $\sigma$-field $\mathcal{B}$ is: What is a countably generated $\sigma$-algebra? Can't find a definition online.
Does this formulation include Markov chains on countable spaces?
Thank you.