Lemma 1.16 -- harmonic functions of an irreducible chain are constant

Let $V$ be a finite type (with decidable equality), and let $P$ be a $V \times V$ matrix with real entries. Assume: (1) $P$ is stochastic in the sense of this bundle's definition, meaning every entry is nonnegative ($0 \le P_{x,y}$ for all $x, y \in V$) and every row sums to one ($\sum_{y \in V} P_{x,y} = 1$ for all $x \in V$); (2) $P$ is irreducible in the sense of this bundle's definition, meaning for every pair $x, y \in V$ there exists a natural number $t \ge 0$ (the value $t = 0$, for which $P^0$ is the identity matrix, is permitted) such that the $(x,y)$ entry of the matrix power $P^t$ is strictly positive: $0 < (P^t)_{x,y}$; and (3) $h : V \to \mathbb{R}$ is harmonic for $P$ in the sense of this bundle's definition, meaning

Then the theorem asserts that for every pair of elements $x, y \in V$ (both universally quantified), $h(x) = h(y)$ — that is, $h$ takes the same value at any two points of $V$. Edge cases silently included: if $V$ is empty the conclusion is vacuously quantified (and moreover the row-sum condition $\sum_y P_{x,y} = 1$ is unsatisfiable over an empty index set only when some $x$ exists, so for empty $V$ all hypotheses hold vacuously and the conclusion is vacuous as well); no aperiodicity, no stationary distribution, and no reversibility assumptions appear in the statement.