Xrfgtjtk

Let ${X_n}_{n=1}^{infty}$ be a equence of random variables defined on $(Omega, mathcal{S}, mathcal{P})$

Let $E = {omega: lim_{n rightarrow infty} X_n(omega)$ exists }$

Let $F = {omega: lim_{n rightarrow infty} frac{1}{n} sum_{i=1}^{n} X_i(omega)$ does not exist $}$

Prove:

Prove that $Ein mathcal{S}$ and $Fin mathcal{S}$

Here are my thoughts. If the sequence is all defined on the same probability space they all must be identically distributed.

If we let $A_n$ = ${omega: X_n(omega) = c$, where c is any real number $}$
and since $X_n$ is a set function that maps to the reals, $A_n$ = $Omega$ for all n, and $lim_{n rightarrow infty} A_n$ = $bigcap_{n=1}^{infty}A_n = E=Omega$

Similarly let $B_n= {omega: frac{X_i(omega)}{n} = c$, where c is any real number $}$.

$bigcap_{i=1}^n B_n subset B_n$, thus $bigcap_{i=1}^n B_n in mathcal{S}$

Because $sigma$-algebras are closed under countable unions:

$bigcup_{n=1}^{infty} bigcap_{i=1}^{n} B_n in mathcal{S} $

And by DeMorgans Laws:

$bigcap_{n=1}^{infty}bigcup_{i=1}^nB_n^c in mathcal{S}$

Note:

$bigcap_{n=1}^{infty}bigcup_{i=1}^nB_n^c$

= $bigcap_{n=1}^{infty}bigcup_{i=1}^n {omega: frac{X_i(omega)}{n} neq c$, where c is any real number $}$

= $bigcap_{n=1}^{infty}{omega: sum_{i=1}^n frac{X_i(omega)}{n} neq c$, where c is any real number $}$

= ${omega: lim_{n rightarrow infty} sum_{i=1}^n frac{X_i(omega)}{n} neq c$, where c is any real number $}$

= $F$

Hence $F in mathcal{S}$

I feel like I used a lot of tricks, and assumptions, but am having trouble finding a better arguement.