Definition (Harris recurrent)

A set $A\subset \mathbb{X}$ is said to be Harris Recurrent if it is visited infinitely often Almost surely, i.e. $$P_{x}(\eta_{A}=\infty)=\mathbb{P}(\{ \eta_{A}=\infty \}\mid x_{0}=x)=1$$

Definition (Harris recurrent chain)

A Markov chain is called Harris recurrent if it is μ-irreducible and every set is .