Royden states that to prove Fatou's lemma it is ncessary and sufficent to show that if $h$ is any bounded measurable function of finite support for which $0\leq h\leq f$ on $E$, then
$$
\int
_E h\leq \lim \inf \int_E f_n$$
Why did Royden choose to construct such a function instead of working directly with $f$?
Best Answer
By definition, $\int f$ is the supremum of $\int h$ for $0 \leq h \leq f$ bounded measurable of finite support. So at the very end, you just need to take the sup over such $h$ to get $\int f \leq \liminf \int f_n$. He uses such $h$ is order to apply the bounded dominated convergence theorem, which he proves earlier. This allows him to prove Fatou in terms of dominated convergence instead of giving a completely independent proof (and potentially repeating a bunch of work).