Can we definition the liminf for a family of sets with continuous parameter and get the corresponding Fatou's lemma? Let $(E_\alpha)_{\alpha\in I}$,where $I$ is an index set (for example, $[0,1]$). Can we define the corresponding liminf, for example, $$\liminf\limits_{\alpha\searrow 0}E_\alpha=\bigcup_{r>0}\bigcap_{0<\alpha<r}E_\alpha.$$
My desired result is the following continuous parameter type Fatou's lemma: Let $\mu$ be a finite meaure and each $E_\alpha$ is measurable, then
$$\mu(\liminf\limits_{\alpha\searrow 0}E_\alpha)\le \liminf\limits_{\alpha\searrow 0}\mu(E_\alpha).$$
I am not sure whether these make sense. Any comments are welcome! Many thanks!
 A: Note that
\begin{align*}
\bigcup_{r>0}\bigcap_{0<\alpha<r}E_{\alpha}=\bigcup_{n=1}^{\infty}\bigcap_{0<\alpha<1/n}E_{\alpha}.
\end{align*}
Indeed, for $r>1$, we have
\begin{align*}
\bigcap_{0<\alpha<r}E_{\alpha}\subseteq\bigcap_{0<\alpha<1}E_{\alpha}\subseteq\bigcup_{n=1}^{\infty}\bigcap_{0<\alpha<1/n}E_{\alpha}.
\end{align*}
Now we let
\begin{align*}
S_{n}=\bigcap_{0<\alpha<1/n}E_{\alpha},
\end{align*}
one sees that $(S_{n})_{n=1}^{\infty}$ is an increasing sequence. Hence,
\begin{align*}
\mu(\liminf_{\alpha}E_{\alpha})=\mu\left(\bigcup_{n=1}^{\infty}S_{n}\right)=\lim_{n}\mu(S_{n}).
\end{align*}
But then for a fixed $n$, we have
\begin{align*}
\mu(S_{n})\leq\mu(E_{\alpha}),\quad 0<\alpha<1/n,
\end{align*}
this allows us to take $\liminf$ both sides with respect to $\alpha\downarrow 0$, which leads to
\begin{align*}
\mu(S_{n})\leq\liminf_{\alpha}\mu(E_{\alpha}).
\end{align*}
Now we take $n\rightarrow\infty$ and note that $\liminf_{\alpha}\mu(E_{\alpha})$ is a constant with respect to $n$ and we are done.
