Every finite state space Markov chain admits an Invariant probability measure.

Assume $S$ is finite and $P$ is irreducible. Then $P$ must be positive recurrent, and hence has a unique invariant distribution. # Countable state space

For an Irreducible Markov chain with countable $\mathbb{X}$, there can be at most one Invariant probability measure.

Let $\alpha \subset \mathbb{X}$ be an atom such that $$E_{\alpha}[T_{\alpha}^{(1)}]<\infty$$ (i.e. positive recurrent) then $\{ X_{i} \}_{i=1}^{\infty}$ admits an invariant probability measure.

Let $\{ X_{i} \}_{i=1}^{\infty}$ be Irreducible, and aperiodic. If $\alpha$ is (n-μ)-small for some $n\in \mathbb{Z}_{+}$ and $$E_{\alpha}[\tau_{\alpha}]<\infty$$ (i.e. positive recurrent) then there exists invariant $\pi$.

Let $\{ X_{i} \}_{i=1}^{\infty}$ be Harris Recurrent. If $\alpha$ is petite and $$E_{\alpha}[\tau_{\alpha}]<\infty$$ (i.e. positive recurrent) then the Markov chain is Positive Harris Recurrent (and $\exists$ unique invariant $\pi$).