I'm just starting with martingales and an exercise for the reader asks that I extend the well-known result $E(\sum_i^N X_i)=E(X_1)E(N)$ for iid $X_i$ to the following case to show the power of martingales.
Let $F_n$ be a filtration, $T$ a finite expectation $F_n$ stopping time and $X_i$ stochastic process with $\sup_i E(\lvert X_i\rvert)<\infty$. Further, assume that $X_i$ is $F_i$ measurable and $X_i\perp F_{i-1}$ as well as $E(X_i)=E(X_1)$. Show $$E\left(\sum_i^TX_i\right)=E(X_1)E(T).$$
For iid $X_i$ and a random variable $T$, this is a typical exercise and I have done that before. In this case however, I am absolutely stumped, as the $X_i$ are not even necessarily independent of $T$!
If I'm not mistaken, the family $(X_i)$ is a martingale, so I believe the intuition here is that it doesn't matter when you stop in a zero drift game, correct? But I don't know how to get that proof-idea (if you can call it that) formalized.