uniform integrability of a martingale related to $E_x(V_A)$ for a hitting time $V_A$

Question

Let $\{X_n\}$ be a Markov chain on a countable state space $S$ s.t. $S-A$ is finite. We denote the hitting time $V_A = \inf \{n \ge 0: X_n \in A\}$. We then consider a function $g$ that satisfies $(*) g(x) = 1 + \sum_y P(x,y)g(y)$ for any $x \not \in A$. In particular $E_x(V_A)$ satisfies $(*)$. I have shown that $Y_n = g(X_{n \wedge V_A}) + n \wedge V_A$ is a martingale and I want to show that it is also u.i. but I am stuck. My first idea was to use $E_x(V_A)$ as a dominating function, but I realized that this would be incorrect since $E_x(V_A)$ is the minimal solution to $(*)$ that is $0$ on $A$. A related result that could be of use is that there is $N, \epsilon >0$ s.t. $P(V_A \ge kN) \le (1-\epsilon)^k$ for all $k$. However, I'm not sure how to use this to prove the uniform integrability of $Y_n$.