Definition (Positive Recurrent)

A set $A\subset \mathbb{X}$ is positive recurrent if $$\begin{align*} E_{x}[\tau_{A}]&<\infty, \ \forall x\in A \end{align*}$$That is, not only w.p.1., the MC will return to $A$ regardless of where it starts within $A$. It requires on average, the expected return time to be finite.