I understand everything in this proof concerning the strong law of large numbers, except for the line highlighted in red. I do not understand why $$\frac{X_1 + …+X_n}{n}$$ is measurable with respect to the $\sigma$-field $\cap_{k=1}^\infty \sigma(X_{k+1}, X_{k+2}, …)$.
My attempt at a justification:
Fix $k \in \mathbb{N}$. $\frac{X_{k+1} + …+X_{k+n}}{n}$ converges almost surely by the identical argument to the case where $k = 0$. It suffices to show that $\text{lim}_{n \rightarrow \infty} \frac{X_{k+1} + …+X_{k+n}}{n} = \text{lim}_{n \rightarrow \infty} \frac{X_1 + …+X_n}{n}$, for in that case, since $\frac{X_{k+1} + …+X_{k+n}}{n}$ is a $\sigma(X_{k+1}, X_{k+2}, …)$ measurable random variable for every $k$, the result follows immediately.
Now fix $N > k$. For all $n \ge N$, we have
$$\left|\frac{X_{k+1} + …+X_{k+n}}{n} – \frac{X_1 + …+X_n}{n} \right| = \left|\frac{X_{n+1} + …+X_{n+k}}{n} – \frac{X_1 + …+X_k}{n} \right|$$
(all the middle terms $X_{k+1}, …, X_{n}$ cancel out)
Now since $\frac{X_1 + …+X_k}{k}$ is a convergent sequence, it is bounded so we can bound the second term inside the last absolute value by dividing and multiplying by $k$, but I don't know how to do the same thing for the first term based on the indices being different. Please help!
For reference, this is taken from Jacod and Protter's "Probability Essentials" textbook.
Best Answer
I just realised the result is probably not that obvious.
Denote $\mathscr{G}_{n} = \sigma(X_k; k \geq n).$ Then, $$X = \lim_{p \to \infty} \dfrac{S_{n + p}}{n + p} = \lim_{p \to \infty} \left( \dfrac{S_n}{n + p} + \dfrac{X_{n + 1} + \ldots + X_{n + p}}{n + p} \right) = \lim_{p \to \infty} \dfrac{X_{n + 1} + \ldots + X_{n + p}}{n + p}$$ is measurable relative to $\mathscr{G}_n,$ thus to $\bigcap\limits_{n \in \mathbf{N}} \mathscr{G}_n,$ which is the tail sigma field (by definition). Q.E.D.