Application of Fatou's Lemma

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Let $\{E_n\}$ be events in a Borel field $\mathscr{F}$, we have $$\mathscr{P}\{\underset{n}{\limsup}E_n\}\geq\underset{n}{\overline{\lim}}\mathscr{P}\{E_n\},$$ $$\mathscr{P}\{\underset{n}{\liminf}E_n\}\leq\underset{n}{\underline{\lim}}\mathscr{P}\{E_n\}.$$



$\bullet$ Proof.

Recall Fatou's Lemma.  If $|X_n|\geq0$ a.e. on $\Lambda$, then $$\int_\Lambda\underset{n\rightarrow\infty}{\liminf}X_n\,d\mathscr{P}\leq\underset{n\rightarrow\infty}{\liminf}\int_\Lambda X_n\,d\mathscr{P}.$$Furthermore, if for all $n$, $|X_n|\leq Y$ a.e. on $\Lambda$ with $\mathscr{E}(Y)<\infty$, the above remains true as well as $$\int_\Lambda\underset{n\rightarrow\infty}{\limsup}X_n\,d\mathscr{P}\geq\underset{n\rightarrow\infty}{\limsup}\int_\Lambda X_n\,d\mathscr{P}.$$See Convergence Theorems

Back to the proof.  We directly apply Fatou's Lemma to show these results.  Define $$X_n(\omega)=\begin{cases}1 & \mbox{, if }\omega\in E_n;\\0 & \mbox{, if }\omega\in E_n^c\end{cases}.$$Clearly, $0\leq X_n\leq 1$ for all $n$ with $\mathscr{E}\{X_n\}=\mathscr{P}\{E_n\}$ and $$\underset{n\rightarrow\infty}{\underline{\lim}}X_n=I\{\underset{n\rightarrow\infty}{\underline{\lim}}E_n\}\mbox{ and }\underset{n\rightarrow\infty}{\overline{\lim}}X_n=I\{\underset{n\rightarrow\infty}{\overline{\lim}}E_n\}.$$Then by Fatou's Lemma, we have
$$\mathscr{P}\{\underset{n}{\liminf}E_n\}
 = \mathscr{E}\{\underset{n\rightarrow\infty}{\underline{\lim}}X_n\}
 \leq\underset{n\rightarrow\infty}{\underline{\lim}}\mathscr{E}\{X_n\}
 = \underset{n}{\underline{\lim}}\mathscr{P}\{E_n\},$$ $$\mathscr{P}\{\underset{n}{\limsup}E_n\}
 = \mathscr{E}\{\underset{n\rightarrow\infty}{\overline{\lim}}X_n\}
 \geq\underset{n\rightarrow\infty}{\overline{\lim}}\mathscr{E}\{X_n\}
 = \underset{n}{\overline{\lim}}\mathscr{P}\{E_n\}.$$

$\Box$