$\psi$ irreducibility and ergodicity of Markov Processes

Question

How is Markov chain splitting technique useful for inferring ergodicity of a Markov Chain?Assume that I am working with general state space (uncountable say $R^{N}$ but time is discrete. I want to show that the Markov Process is ergodic. I guess that it suffices to show that it is Harris recurrent. To show Harris recurrence I guess that it suffices to show there exists an atom (obtained via splitting the chain after using minorization criteria) the return time (or hitting time) to which has finite mean.