# Fatou's lemma - Royden's proof

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 $$f$$?

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).
• I just want to follow up on this a bit further. You are taking the supremum of the set of functions $h$ that are bounded by $f$ right? Mar 26, 2021 at 2:20
• Instead of taking the supremum over a uncountable set of functions, can we prove it by contradiction in the end by assuming $\int f>\lim \inf f_n$ and use $\int_E h\leq\lim \inf \int_E f_n$ to draw the conclusion. Because I am not sure how the supremum over a uncountable set of functions works. Mar 26, 2021 at 2:45
You can also let $$g_n=inf_{i \geq n} \{f_n\}$$ and note that $$\int g_n \leq \int f_n$$ for each n. Moreover $$g_n$$ increases monotonically to lim inf $$f_n$$ . Hence by monotone convergence theorem $$\int g_n = \int lim inf f_n$$. Since $$\int g_n \leq \int f_n$$ we have $$\int lim inf f_n \leq lim inf \int f_n$$. The final step is to note that pointwise convergence implies lim inf $$f_n$$ = f