Stuck in a place in the proof of Fatou's lemma I stuck in a step in the proof of Fatou's lemma. The books is Royden's. I don't know what the red line said. Isn't $\int_E h_n\le\int_E f_n$ deduced from the Monotonicity of Lebesgue integrals? And where did the second red line come from? Is it simply both take the operator $\liminf$?


 A: The monotonicity of the lebesgue integral is used on the first read line.
Now for the second line note since $\int_E f_n \geq \int_Eh_n,\forall n \in \Bbb{N}$
then $\liminf_n \int_Ef_n \geq \liminf_n \int_E h_n=\lim_n \int_E h_n$
A: The definition of the integral of  positive functions in Rudin is $$\int_E f = \sup \left\{ \int_E h : h\leq f , \text{ and the function } h \text{ is bounded with bound support } \right\} $$
We say $h$ is of bounded support if $h\ne 0 $ on a set with finite measure. By this definition we are connected with the definition of integral of bounded functions on bounded sets  which is connected to the definition of simple functions. 
So to prove Fatus we look at such $h's$ bounded with bounded supports if we can show that for any bounded function $h$ with bounded support and  $h \leq f $ with  $\lim h \leq \lim \inf \int f_n $ then we can conclude $\int f \leq \lim \inf \int f_n $. 
When we proved that for any $h$ with the conditions we get $\int h = \lim \int h_n \leq \lim \inf \int f_n $  Then by the definition of $\int f$  for any epsilon $\epsilon >0 $ there exists $h$ bounded with bounded support such that $\int_E f < \int_E h + \epsilon $ then the Lemma follows  
