Uniqueness of the stationary law for an irreducible positive-recurrent chain

Existence: since $p$ is irreducible and positive recurrent, [F1] supplies an invariant probability $\pi$ for $p$.

Uniqueness: let $\pi$ and $\rho$ be invariant probabilities. Fix $x\in E$; the Kac identity [F2] applied to $\pi$ gives $\pi (x)=1\mathrm{/}{\mathbb{E}}_x{T}_x^{+}$, and applied to $\rho$ gives $\rho (x)=1\mathrm{/}{\mathbb{E}}_x{T}_x^{+}$, the denominator being the same positive finite number because ${\mathbb{E}}_x{T}_x^{+}$ depends only on the chain and the state. Hence $\pi (x)=\rho (x)$ for every $x\in E$.

Two probability measures on the countable discrete space $E$ with equal singleton masses are equal: by [F3] both assign to every $E^{\mathrm{\prime }}\subseteq E$ the value ${\sum }_{x\in E^{\mathrm{\prime }}}\pi (\{x\})$, the same series. Hence $\pi =\rho$, and the invariant probability is unique.

Combining steps 1.1 and 2.1, $p$ has exactly one invariant probability, and step 1.2 exhibits it as $\pi (x)=1\mathrm{/}{\mathbb{E}}_x{T}_x^{+}$.

Boundary and axiom cases: aperiodicity is never used, so the corollary covers periodic positive-recurrent chains; a reducible chain may have many invariant probabilities, such as the identity matrix on two states where every mixture of the two absorbing laws is invariant, and the irreducibility hypothesis is used in [F1] and in the positivity statement of [F2]; a null-recurrent chain has no invariant probability at all by [F1], so uniqueness is then vacuous rather than false; an empty state space carries no probability law; the equality $\pi =\rho$ is checked at every singleton, which is exactly the determined family of [F3]; and AC [A1] enters only through the chain-law suppliers of [F1] and [F2].