Can you recommend good lecture notes (or a book) about this topic? Basically I would like something which covers more or less the first chapter of the book "Markov Processes" by Ethier and Kurtz, but hopefully written more like lecture notes (a bit more verbose and more examples, ideally with connections to probability) and less like in a reference book. I would prefer if it were available online, but it is not strictly necessary. Thanks!
A side question - generally my knowledge of functional analysis is rather lacking and I would probably benefit a lot from improving it. Can you recommend a good book which goes more or less from the basics (I had an introductory functional analysis lecture some years ago, but I could revisit that again and then move on). I am mostly concerned about those parts of functional analysis that have connections to stochastic processes, but I think I generally need a good basic overview (a bit less rudimentary than the one I have now). Again, I prefer verbose style to reference book style, lecture notes that are seriously meant to be read from cover to cover would be ideal.