I don't understand what a stopped martingale really is, $(M_{T\wedge n})$. In my book it is defined as.
Let $(M_n)_{n \in \mathbb{N_0}}$ be a martingale with respect to $(F_n)_{n \in \mathbb{N_0}}$ and let $T$ be a stopping time with respect to $(F_n)_{n \in \mathbb{N_0}}$. Define
$$ (M_{T\wedge n}): \Omega \to \mathbb{R}, \ \omega \to M_{T(\omega) \wedge n}(w) $$
I think there is something important in the notation $T \wedge n$ that I have missed.
$T \wedge n = \min\{T,n\}$ so this stopped martingale just keeps constant value after stopping time $T$ actually occurs.
This is so even though the actual process $M_n$ may continue evolving.
For a simple discrete example, let $M_n$ be the position of the regular unbiased random walk starting at $0$. Let $T$ be the 2nd time that $M$ hits $1$. One possible path $M_n(\omega)$ would be
0, -1, -2, -1, 0, 1, 0, -1, 0, 1, 2, 3, 2, ...
On the same path, the stopped martingale $M_{n\wedge T(\omega)} (\omega)$ would look like
0, -1, -2, -1, 0, 1, 0, -1, 0, 1, 1, 1, 1, ...