Thank you very much for all these explanations. I will go to look into these references

Thank you for your answer. What do you mean by "irreducibility" of the state space in the general result that you quote in the first paragraph? In the book Markov chain and stochastic stability, the ergodicity of Markov chain is proven under the assumption that the chain is positive Harris, i.e. $\psi$-irreducible, admitting an invariant probability measure and Harris recurrent. Do you mean Harris recurrent is not necessary or your definition of irreducibility includes Harris recurrence? Could you give me a reference of this general result?

@RW I've edited my question. Hope it's more clear now.