Reddit reviews General Irreducible Markov Chains (Cambridge Tracts in Mathematics)
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By old-fashioned definitions. See General Irreducible Markov Chains and Non-Negative Operators by Nummelin or Markov Chains and Stochastic Stability by Meyn and Tweedie.
Markov chains used to be defined the way you say, because there were no good methods to handle general state spaces. Now there are, so the old-fashioned definition leaves out the most interesting applications for no good reason.