Here is the theorem and proof presented in Probability and Random Processes (Grimmett).
Theorem: Let $(X, \pmb{\mathscr{F}})$ be a martingale and $\tau$ a stopping time. If $(i)$ $\tau$ is finite, i.e. $\mathbb{P}(\tau < \infty) = 1$ $(ii)$ $\mathbb{E}[|X_{\tau}|] < \infty$, and $(iii)$ $\lim_{n \to \infty}\mathbb{E}[X_n \mathbb{I}_{\{\tau > n\}}]=0$, then we have that the martingale property is preserved under random stopping. That is, $\mathbb{E}[X_\tau] = \mathbb{E}[X_0]$.
Proof: It can be observed that $X_\tau = X_{\tau \wedge n} + (X_\tau - X_n)\cdot \mathbb{I}_{\{\tau > n\}}$. Since $\tau \wedge n$ is a bounded stopping time, we know that the martingale property is preserved. Therefore, $\mathbb{E}[X_\tau] =\mathbb{E}[X_0] + \mathbb{E}[X_\tau\cdot \mathbb{I}_{\{\tau > n\}}] - \mathbb{E}[X_n\cdot \mathbb{I}_{\{\tau > n\}}].$ Notice,
- $\lim_{n \to \infty}\mathbb{E}[X_n\cdot \mathbb{I}_{\{\tau > n\}}] = 0$ by $(iii)$.
- $\mathbb{E}[X_\tau\cdot \mathbb{I}_{\{\tau > n\}}] = \sum_{k=n+1}^{\infty} \mathbb{E}[X_k \cdot \mathbb{I}_{\{\tau = k\}}]$ which is the tail of a convergent series by $(ii)$. Therefore, we know that $\lim_{n \to \infty} \sum_{k=n+1}^{\infty} \mathbb{E}[X_k \cdot \mathbb{I}_{\{\tau = k\}}] = 0$
We have that $\mathbb{E}[X_\tau] = \mathbb{E}[X_0]$ as $n \to \infty$.
Question: I understand how parts $(ii)$ and $(iii)$ are being used to prove this theorem; however, I can't seem to figure out why we have part $(i)$ as another condition. Does anyone know how part $(i)$ comes into play?
Without (i) the second condition does not even make sense. We are not assuming that the martingale is convergent. So what Is $X_{\tau}$ in (ii)?. It is defined only when $\tau <\infty$.