I read a book that claims the following: Let $\{X_{n} : n\in\mathbb{N}\}$ be a martingale and $T$ a stopping time. Then \begin{equation} E[X_T 1_{\{T \leq n\}}] = \sum_{j=0}^{n} E[X_j 1_{\{T = j\}}] = \sum_{j=0}^{n} E[X_0 1_{\{T = j\}}] = E[X_0 1_{\{T \leq n\}}] \end{equation}
I don't see the second equality. And yes, it is really $X_T 1_{\{T \leq n\}}$ and not $X_{T \wedge n}$.
This should be a counter example: For the standard integer random walk with $X_0 = 0$ a.s., $n=1$ and $T = \inf\{n : X_n = 1\}$ \begin{equation} E[X_T 1_{\{T \leq n\}}] = P(T=1) = 1/2 > 0 = E[X_0 1_{\{T \leq n\}}] \end{equation}
Still I'm a bit confused. Is there something wrong with my example?
Thanks for your help.