Breaking down generalized Fatou's Lemma into steps

Question

If $\{f_n\}$ be a sequence of non-negative measurable functions, then $\int_E \lim\inf f_n \leq \lim\inf \int_E f_n$.

I need for this to either be broken down into steps or for a hint to be provided. If this is a large proof, please break it down into steps. If there is an easy proof, provide a hint. I would prefer it if nobody posts a complete solution.

Answer 1

Define new functions

$g_k(x) = \inf_{n\geq k} f_n (x)$

Then, by definitnion you have that $\liminf f_n = \lim g_n$.

Now apply (the non-generalized) Fatou's Lemma to the functions $g_n$. I provide a full solution (Hidden below)

Let $g(x)$ denote the limit of $g_n(x)$ (which exists because $g_n(x)$ is monotone) then by the standard Fatou's Lemma $$\int \lim g_n(x) dx =\int\liminf f_n(x)dx\leq \liminf \int g_n(x) dx$$ Moreover since $g_k(x) = \inf_{n\geq k} f_n(x)$ we always have that $g_k(x)\leq f_k(x)$. Insert this to the right hand side you will complete the proof.