understand proof of strong law of large numbers (using reverse martingale)

Question

I am reading the proof of strong law of large numbers using reverse martingale here. There are several points about conditional expectation that I do not understand. They all seem pretty intuitive but I want someone to explain in more details rigorously.

1. Why does $E[X_i|Y_n, X_{n + 1}, X_{n + 2}, \cdots] = E[X_i|Y_n]$ for any $i \leq n$? This looks obvious since $X_i$ is independent of $X_{n + 1}, X_{n + 2}, \cdots$, but is there any theoretical result behind the claim?

2. How do we know $E[X_i|Y_n] = E[X_j|Y_n]$ for any $i ,j \leq n$ "by symmetry"?

3. Why does $E[X_{n + 1}|Y_{n + 1}] = \frac{Y_{n + 1}}{n + 1}$? I can still see the result intuitively, but is there a way to approach such problems systematically?

For notations, $X_k$s are the i.i.d. random variables with finite mean given in the theorem, $Y_n$ is the sum of $X_1$ to $X_n$.

Answer 1

1. If $X$ is a real-valued random variable, either nonnegative or integrable, and $\mathcal H$ and $\mathcal G$ are two $\sigma$-algebras such that $\mathcal G$ is independent of $\sigma(\sigma(X),\mathcal H)$, then $\mathbb E[X\vert\sigma(\mathcal H,\mathcal G)]=\mathbb E[X\vert\mathcal H]$ almost surely. Apply this with $X=X_i$, $\mathcal H=\sigma(Y_n)$ and $\mathcal G=\sigma(X_k,k\ge n+1)$.
2. If $X$ and $X'$ are real-valued random variables, either nonnegative or integrable, and $Y$ is a random variable such that $(X,Y)$ and $(X',Y)$ have the same distribution, then $\mathbb E[X\vert Y]=\mathbb E[X'\vert Y]$ almost surely. Apply this with $X=X_i$, $X'=X_j$ and $Y=Y_n$, as by symmetry $(X_i,Y_n)$ and $(X_j,Y_n)$ have the same distribution.
3. Using bullet 2 we have that $$ (n+1)\mathbb E[X_{n+1}\vert Y_{n+1}]=\sum_{k=0}^{n+1}\mathbb E[X_k\vert Y_{n+1}]=\mathbb E\left[\sum_{k=0}^{n+1}X_k\vert Y_{n+1}\right]=\mathbb E[Y_{n+1}\vert Y_{n+1}]=Y_{n+1}. $$