Let $X_1, \dots, X_n$ i.i.d random variables. Put $S_n:= \sum_{k=1}^n X_k$.
Show that $\mathbb{E}[X_1 I_{\{S_n \in A\}}]= \mathbb{E}[X_j I_{\{S_n \in A\}}]$ for $1 \leq j \leq n$, where $A$ is an arbitrary Borel set.
A possible idea of mine was to prove that the random vectors $(X_1, S_n)$ and $(X_j, S_n)$ have the same joint distribution. The statement intuitively looks obvious because the sum $S_n$ is symmetric in all variables $X_1, \dots, X_n$ so it shouldn't matter which variable we write on the front.
How can I formally show this? (Fubini and all these theorems like are allowed). These things are always hard to show yet seem so intuitive.
The i.i.d. random variables have this property: Let $\pi$ be any permutation of $\{1,\dots,n\}$. Then the two vectors $$ (X_1,X_2,\dots,X_n)\qquad \text{and}\qquad (X_{\pi(1)},X_{\pi(2)},\dots,X_{\pi(n)}) $$ have the same joint distribution. This is called exchangeable. See that wiki site for more information.