Following the book about Markov Processes from Ethier and Kurtz I understood that the most general requirements for the infinitesimal generator point of view for defining a Markov process from a given operator (which satisfies some hypothesis) in some space $E$, is that this space must be locally compact and separable. My question arises after reading some articles where the space used is only a Polish space, which is the usual setting for probabilities, and I would like to know how can i apply this theory in the given context. What happen if I'm missing the locally compact requisite?
Any enlightenment will be appreciated.