Why is there Inequality in Fatou's Lemma?

Question

I'm studying measure theory for the first time, and I just came across Fatou's Lemma.

Why isn't it true that for any sequence of functions $\left\{ f_n \right\}$ in $L^+$ we always have that $$\int \displaystyle \liminf_{n\rightarrow \infty} f_n d\mu =\liminf_{n\rightarrow \infty} \int f_n d\mu\ ?$$

Answer 1

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 $

Answer 2

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.

Actually, taking $f_{2n}=g$ and $f_{2n+1}=h$ would give $$ \int \min\{g,h\}=\min\left\{\int g,\int h \right\}, $$ which cannot hold, for example when $g$ and $h$ are indicator functions of disjoint sets of positive measure.