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 $$\int \displaystyle \liminf_{n\rightarrow \infty} f_n d\mu =\liminf_{n\rightarrow \infty} \int f_n d\mu\ ?$$
Best Answer
There could be several reasons for which the equality does not hold:
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.