Can anyone shed some light on this theorem? Not sure of the significance of the result or how one would apply it.
If $M_n$ is a supermartingale with respect to $X_n$ and $T$ is a stopping time then the stopped process $M_{T \wedge n}$ is a supermartingale with respect to $X_n$. In particular, $\mathrm EM_{T \wedge n} \leq M_0$.