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I am new on martingales and I try to understand the following example:

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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!

Frederick Manfred
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    In what the fact that $S_n$ is $F_n-$measurable help to prove that $\mathbb E[S_n|F_{n-1}]=S_{n-1}$ ? For your last question, it comes from the fact that $X_n$ and $F_{n-1}$ are independent. – Surb Sep 02 '21 at 09:05
  • Okay but then a new question: Why we have that $\mathbb E(S_{n-1}|F_{n-1})=S_{n-1}$? and also why are they independent ($X_n$ and $F_{n-1}$)? – Frederick Manfred Sep 02 '21 at 09:08
  • I think you should start from measure based probability theory, then you will know the meaning of $\mathcal{F}_{n-1}$ and of the word independence. – Q9y5 Sep 02 '21 at 09:12
  • Okay the independence I could solve myself but the first question is still open.. - EDIT: It is $F_{n-1} measurable... Alright, thanks for your help! – Frederick Manfred Sep 02 '21 at 09:14

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