Fatou's theorem

Question

I don't understand how does $(77)$ follow from $(79)$ and $(81)$?

Any help would be appreciated.

Answer 1

Since $g_n\leq f_n$, we have that $\int g_n d\mu\leq \int f_n d\mu$, and hence $$ \liminf_n \int g_n d\mu \leq \liminf_n \int f_n d\mu. $$ Since $g_n$ is an increasing sequence, so is $\int g_n d\mu$, and hence $$ \inf_{i\geq n}\int g_i d\mu=\int g_n d\mu, $$ which means that $$ \liminf_n \int g_n d\mu=\lim_n \int g_n d\mu\stackrel{(81)}{=}\int f d\mu. $$

Answer 2

Because as $n$ gets larger you are taking the infimum on a smaller set.