Why is there Inequality in Fatou's Lemma?

There could be several reasons for which the equality does not hold:

• if the measure space has infinite measure, mass can "escape", for example with $f_n:=\chi_{(n,n+1)}$ on the real line;
• in the case of a finite measure space, there could be a huge diminution of the measure of the support of $f_n$ as $n$ goes to infinity;
• oscillation of the function, for example $f_n(x):=\sin^2(n\pi x)$ on the unit interval.

It is worth mentioning that if $\sup_nf_n$ is integrable, then equality holds (dominated convergence theorem).

consider the sequence of functions$$f_n = n 1_{(0,\frac{1}{n}]} $$

Notice

$$ \int f_n = 1 \implies \lim \int f_n = 1 \implies \liminf \int f_n = 1$$

But, $\lim f_n = 0 \implies \liminf f_n = 0 \implies \int (\liminf f_n ) = 0 $