Let $(X_n)_{n\ge 0}$ be a Markov Chain with finite state space $E \subset \mathbb{R}$. Suppose that the chain has two absorbent states $A = \{i,j\}$ and the other states are transient. Let $g: \mathbb{R} \to \mathbb{R}$ such that $(g(X_n))_{n \ge 0}$ is martingale.
Prove that the absorption probabilities $\rho_{ki}, \rho_{kj}$ where $k \in E \setminus A$ satisfy the system of equations: $$ \begin{split} g(k) &= g(i) \rho_{ki} &+ g(j)\rho_{kj}\\ 1 &= \rho_{ki} &+ \rho_{kj} \end{split} $$ for all $k \in E \setminus A$.