I am new on martingales and I try to understand the following example:
Why we use there that $S_{n-1}$ is $\mathcal{F}_{n-1}$ measurable and not directly that $S_n$ is $\mathcal{F}_n$ measurable? Furthermore how can we conclude that $\mathbb E(X_n\mid \mathcal{F}_{n-1})=\mathbb E (X_n)$? Thanks for some help!