# Inequalities like in Fatou's Lemma

If $\{f_n\}$ is a sequence of non-negative Lebesgue measurable function on $\mathbb{R}$, $f_n\longrightarrow f$ a.e., then

$\textbf{(1.) Fatous Lemma}:$ $\int \lim\inf f_n\leq \lim\inf \int f_n$.

I want to see the importance of limit and infimum in the inequalities. Therefore, I would like to consider following natural questions:

With the same hypothesis, do we have the following inequalities?

$\textbf{(2.)}$ $\int \lim f_n \leq \lim\int f_n$

$\textbf{(3.)}$ $\int \inf f_n \leq \inf\int f_n$

Is the conclusion (1) stronger than (2) and (3)? If yes, how? (any example?)

(2.) holds provided the $\lim f_n$ and $\lim \int f_n$ exist, because in this case they are equal to $\liminf f_n$ and $\liminf\int f_n$.
(3.) also holds since $\inf f_n\le f_n$, so $\int \inf f_n\le \int f_n$ for all $n$, implying $\int \inf f_n\le \inf\int f_n$ (when $t\le s$ for all $s\in S$, then $t\le \inf S)$.