Proving that $E[X_i|S_n ] = \frac{S_n}{n}$ for iid $X_i$ We have $X_1, \dots, X_n$ iid random variables with finite expectations. Defining $S_n = \sum_{i=1}^{n} X_i$, we want to show that $$E[X_i|S_n ] = \frac{S_n}{n}. \quad \forall 1 \leq i \leq n$$
Presently, I'm going with: 
$E[X_i|S_n ] = E[X_j|S_n]$ for all $1 \leq i,j \leq n$ because of identical distributions. Thus$$S_n = \sum_{i=1}^{n} E[X_i|S_n] = n E[X_i|S_n] \implies E[X_i|S_n] = \frac{S_n}{n}.$$
However, I feel I should probably be more formal/rigorous in the $E[X_i|S_n ] = E[X_j|S_n]$ step but am unable to find a way of expressing so. Would appreciate some help. 
Also, in the same setup, in the notes I'm reading, I came across this statement I haven't been able to work out: $\sigma(S_m) \subset \sigma(X_1, \dots X_m)$. Shouldn't there be an equality? 
 A: 
However, I feel I should probably be more formal/rigorous in the $E[X_i|S_n ] = E[X_j|S_n]$ step but am unable to find a way of expressing so. Would appreciate some help. 

The symmetry argument is okay.   The terms in sequence $(X_i)_{i=1}^n$ are iid and their coefficents in the series $S_n$ are identically $1$, so they are interchangeable.  $\mathsf E(X_i\mid S_n)=\mathsf E(X_j\mid S_n)$ for all integer $i,j: (1\leq i{,}j\leq n)$
$$\mathsf E(X_j\mid S_n) {= \tfrac 1n\sum_{i=1}^n\mathsf E(X_i\mid S_n) \\ = \tfrac 1n \mathsf E(S_n\mid S_n) \\ = \tfrac 1n S_n }$$

Also, in the same setup, in the notes I'm reading, I came across this statement I haven't been able to work out: $\sigma(S_m) \subset \sigma(X_1, \dots X_m)$. Shouldn't there be an equality? 

Take, for example, Benoulli random variables $X_1, X_2$ occuring in a sample space represented by outcomes that are the pair of their results.  Ie: $\Omega=\{0,1\}^2$   Then consider the sigma-algebra generated by the atoms.
Then $~~~\sigma(X_1,X_2)= \sigma\big\{\{(0,0)\},\{(0,1)\},\{(1,0)\},\{(1,1)\} \big\} = \mathcal P(\Omega)\\ \sigma(X_1+X_2)=\sigma\big\{\{(0,0)\},\{(0,1),(1,0)\},\{(1,1)\}\big\} \subsetneq \mathcal P(\Omega)$ 
So we have at least one example where there is not equality.   
