Corollary 1.17 (uniqueness) -- at most one stationary distribution

Let $V$ be an arbitrary finite type (possibly empty) with decidable equality, and let $P$ be a $V \times V$ matrix with real entries. Assume: (i) $P$ is "stochastic" in the sense that every entry is nonnegative, $P_{x,y} \ge 0$ for all $x, y \in V$, and every row sums to one, $\sum_{y \in V} P_{x,y} = 1$ for all $x \in V$; (ii) $P$ is "irreducible" in the sense that for every ordered pair $x, y \in V$ there exists a natural number $t \ge 0$ with $(P^t)_{x,y} > 0$ — note that $t = 0$ is allowed, and since $P^0$ is the identity matrix, the condition is automatically satisfied whenever $x = y$, so this hypothesis only constrains pairs with $x \ne y$; (iii) $\pi, \pi' : V \to \mathbb{R}$ are two functions each of which is "stationary" for $P$, meaning that each is a probability distribution on $V$ (all values nonnegative and $\sum_{x \in V} \pi(x) = 1$, likewise for $\pi'$) and each is a fixed point of $P$ acting on row vectors from the left, i.e. $\sum_{x \in V} \pi(x) P_{x,y} = \pi(y)$ for every $y \in V$, and likewise for $\pi'$. The conclusion is that $\pi = \pi'$ as functions on $V$, i.e. $\pi(x) = \pi'(x)$ for every $x \in V$. (If $V$ is empty, the normalization $\sum_x \pi(x) = 1$ reads $0 = 1$, so the stationarity hypotheses are unsatisfiable and the statement holds vacuously; no aperiodicity assumption appears anywhere in the statement.)