TITLE: Can Markov Chain state space be continuous?
QUESTION [9 upvotes]: I looked for a formal definition of Markov chain and was confused that all definitions I found restrict chain's state space to be countable. I don't understand purpose of such a restriction and I have feeling that it does not make any sense. 
So my question is: can state space of a Markov chain be continuum? And if not then why?
Thanks in advance.

REPLY [7 votes]: Yes, it can. In some quarters the "chain" in Markov chain refers to the discreteness of the time parameter. (A notable exception is the work of K.L. Chung.)  The evolution in time of a Markov chain $(X_0,X_1,X_2,\ldots)$ taking values in a measurable state space $(E, {\mathcal E})$ is governed by a one-step transition kernel $P(x,A)$, $x\in E$, $A\in{\mathcal E}$:
$$
{\bf P}[ X_{n+1}\in A|X_0,X_1,\ldots,X_n] = P(X_n,A).
$$
Two fine references for the subject are Markov Chains by D. Revuz and Markov Chains and Stochastic Stability by S. Meyn and R. Tweedie.