The questions asks to $ E[X_1|S_n]$ where $ S_n = \sum_{[n]} X_i $ with $X_i$ i.i.d. of finite expectation. My attempt was to consider an arbitrary Borel set, pull it back under $ S_n $ to get a set in $\sigma(S_n)$, and then average $X_1$ over that pull back and normalize:
$$ E[X_1|S_n] = E[X_1 1_{S_n^{-1}(B)}]/P[S^{-1}_n(B)]$$
for any Borel set $B$.
However, it just doesn't feel right - it seems just symbol pushing, without giving any intuition to what's happening.
Can anyone explain if this is correct, of if I missing something deep?
Hint: The distributions of $(X_1,S_n)$ and $(X_k,S_n)$ coincide hence $E(X_1\mid S_n)=E(X_k\mid S_n)$ for every $1\leqslant k\leqslant n$ (this uses the fact that conditional expectations only depend on joint distributions). Summing these over $k$, one gets $$n\,E(X_1\mid S_n)=\sum_{k=1}^nE(X_k\mid S_n)=E\left(\left.\sum_{k=1}^nX_k\right| S_n\right)=\cdots$$