Let $\{X_n\}$ be a sequence of i.i.d. random variables, we have $$\frac{S_n}{n}\mbox{ converges a.e. }\implies\mathscr{E}|X_1|<\infty.$$
$\bullet$ Proof.
WLOG, suppose $S_n/n\rightarrow0$ a.e., then $X_n/n\rightarrow0$ a.e., that is, $$\mathscr{P}\{|X_n|>n\mbox{ i.o.}\}=0.$$Since $\{X_n\}$ are i.i.d., by the converse of the second Borel-Cantelli Lemma, we have $$\sum_n\mathscr{P}\{|X_n|>n\}<\infty,$$and hence $\mathscr{E}|X_1|<\infty.$
$\Box$
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