I have the following question regarding independence of generated $\sigma$-algebras:
Let $(X_i)_{i\geq 1}$ an i.i.d sequence in $\mathcal{L}^1$, define $S_n=\sum_{i=1}^n X_i$ I am trying to prove that $E(X_1|\mathcal{G} _n)=E(X_1|S_n)$ where $\mathcal{G}_n:=\sigma(\cup_{k=n}^\infty S_k)$
For that purpose, I am wondering if it is true that $\sigma(S_n)$ and $\sigma(\cup \sigma(X_1), \sigma(S_n))$ are independent of $\sigma(\cup_{i=n+1}^\infty S_{i})$.
No: this would imply that $S_n$ is independent of $S_{n+1}$. If you assume integrability, this implies that $$\mathbb E\left[S_{n+1}\right]=\mathbb E\left[S_{n+1}\mid S_n\right]=S_n+\mathbb E\left[X_{n+1}\mid S_n\right]=S_n+\mathbb E\left[X_{n+1}\right]$$ hence $S_n$ is constant.