My professor gave me a very unclear proof of this theorem. It was so messy and unclear, I was unable to write down all the details of the proof.
Theorem: Suppose $\tau \in T$, where $T$ is the set of all stopping times. Then $E[M_{\tau}] = E[M_{0}]$, where $M_n$ is a discrete time martingale.
What I wrote down in my lecture:
Define $N_{*} = max(k: k \in T)$. Write $\tau = \sum_{k=1}^{N_{*}}k I_{\tau=k}$. Then $$E[M_{\tau}] = E[\sum_{k=1}^{N_{*}}M_k I_{\tau=k}] = \sum_{k=1}^{N_{*}}E[M_k I_{\tau=k}]$$
That's all I wrote down. How can I finish the proof?
Earlier in the lecture, we proved that $\forall m < n$
$$E[M_n | F_m] = M_m$$
I'm thinking that if we set $m=0$ we can maybe use this:
$$E[M_n | F_0] = M_0$$
But I'm honestly not sure what to do here.