Question regarding specific steps of Fatou's lemma proof.

Question

I have a question regarding to the following two steps of the proof.

1. How can we get $\int g_n \leq \int f_n$, or $g_n \leq f_n$ for all n, Since by definition of liminf, $g_n \leq f_m $ for all $m \geq n$. Where did the $m$ go?

2. Why did $\int f_n$ become $\liminf\int f_n$ suddenly?

Answer 1

For your first question, since $g_n \leq f_m$ for all $m \geq n$, then, if $m = n\implies m=n \geq n$, so $g_n \leq f_n$.

The second question, what happened was that he passed the liminf in the inequality: $$\int g_nd\mu \leq \int f_n d\mu \implies lim\inf_n \int g_nd\mu \leq lim\inf_n\int f_n d\mu $$ Now, since $g_n$ is increasing, then $$lim\inf_n \int g_nd\mu =lim_n \int g_nd\mu $$ Therefore, you get $$lim_n \int g_nd\mu \leq lim\inf_n\int f_n d\mu $$

Answer 2

Just write the set $\{f_i : n\leq i < \infty\}$ explicitly, i.e. $\{f_n,f_{n+1},...\} \forall n\geq 1$.

Then you take infimum over this set, i.e $g_n$

So, it is obvious that $g_n\leq f_n\ \forall n\geq 1$ , then take the integral both side, we get $$ \int g_n d\mu\leq \int f_n d\mu.............(*) $$

Now concentrate on $g_n$.

$g_n \leq g_{n+1}$ and $\lim g_n = \liminf\{f_i : n\leq i < \infty\}= \liminf f_n = f$, this is from definition of limit infimum of sequence.

Then use Monotone convergence theorem on ${g_n}$. We get $$\lim \int g_n d\mu = \int f d\mu \\ \Rightarrow \liminf\int g_n d\mu = \int f d\mu $$ since $\lim \int g_n$ exists and $\lim \int g_n = \liminf\int g_n d\mu$.

now take $\liminf$ both side of (*) you get the desired result.