$(X_n)_n$ is a sequence of independent random variable, such that $E[X_n]=0.$ Let $Y \in L^2$ such that for all $n \in \mathbb{N}^*,Y-\sum_{k=1}^nX_k$ is independent of $(X_1,...,X_n).$
Prove that for every $n \in \mathbb{N}^*,X_n \in L^2$ and that $\sum_{k=1}^nX_k$ converges a.s.
Since $Y=Y-\sum_{k=1}^nX_k+\sum_{k=1}^nX_k \in L^2,$ by independence $X_n \in L^2.$ To prove that $\sum_{k=1}^nX_k$ converges a.s, it's sufficient to prove the convergence of $\sum_nVar(X_n),$ also we can derive from above, that $\sum_{k=1}^nVar(X_k)=\sum_{k=1}^nE[X_kY].$
Any suggestions, hints how to continue?
If $Y=Z+W$ where $Y \in L^2$ and $Z,W$ are independent, then $Z,W \in L^2$ and consequently Var$(Y)=$Var$(Z)+$Var$(W)$. The first assertion follows from Fubini's Theorem: We may assume $Z,W$ have mean $0$. Writing $\mu, \nu$ for the laws of $Z,W$ respectively, we have $$E(Y^2)=\int \int (z+w)^2 \, du(z)\, d\nu(w)$$ so $Z+w \in L^2$ for $\nu$-almost all $w$, whence $Z \in L^2$. In the question we get Var$(X_1+...+X_n) \le$Var$(Y)$.