See Problem below. How did they come up with the expression:
$$M_T = \sum \limits_{j=0}^{k} M_j I_{[T=j]}$$
All they told me for the problem statement was that $\{M_n, n \ge 0\}$ is a stochastic process, I'm not understanding why they suddenly decided to add the sequence together to get $M_T$, why would it be like that? I think they left out part of the problem statement.
The stopping time is a partition of the sample space $\Omega$ and since it is bounded we have that: $$ \Omega = \bigcup_{j=0}^k \{ T = j \} $$
So, $$ 1 = \sum_{j=0}^k I_{ \{T=j\} } $$
And therefore: $$ M_T = M_T\times 1 = \sum_{j=0}^k M_TI_{ \{T=j\} } = \sum_{j=0}^k M_jI_{ \{T=j\} } $$