Show that $ \int \liminf f_n \leq \liminf \int f_n \leq \limsup \int f_n \leq \int \limsup f_n$ 
Let $g$ be a non-negative integrable function over $E$ and suppose $\{f_n\}$ is a sequence of measurable functions on $E$ such that for each $n$, $|f_n| \leq g$ a.e. on $E$. Show that 
  $$ \int \liminf f_n \leq  \liminf \int f_n \leq \limsup \int f_n \leq \int \limsup f_n.$$

I know that this problem is an application of the Lebesgue dominated convergence theorem. 
Any idea of how to go about it thanks, I am really having a hard time with this problem.
 A: By possibly excising a set of measure 0 we can assume that $|f_n| \leq g$ holds on $E$.\
Let $$g_n =\inf\limits_{k\geq n} f_k \leq f_n \ then \ g_n \rightarrow \lim\limits_{n \to \infty} \inf f_n$$
Note that from $-g \leq |f_n| \leq g$ for all $n$ it also follows that $-g \leq g_n \leq g$ for all $n$ and thus $|g_n| \leq g$\
Using LDCT it follows that 
$$\int\limits_{E} \lim\limits_{n \to \infty} \inf f_n = \lim\limits_{n \to \infty} \int\limits_{E} g_n = \lim\limits_{n \to \infty} inf \int\limits_{E} g_n \leq \lim\limits_{n \to \infty} \int\limits_{E} f_n$$\
Also
$$\int\limits_{E} \lim\limits_{n \to \infty} inf f_n  \leq \lim\limits_{n \to \infty} \int\limits_{E} f_n \leq \int\limits_{E} \lim\limits_{n \to \infty} \sup f_n  \  \  (*)$$
Similarly,Let $h_n =\sup\limits_{k \geq n} f_k \geq f_n$ then $h_n \rightarrow \lim\limits_{n \to \infty} \sup f_n$ and note that $|h_n| \leq g$\
Again, using LDCT we get
$$\int\limits_{E} \lim\limits_{n \to \infty} \sup f_n = \lim\limits_{n \to \infty} \int\limits_{E} h_n = \lim\limits_{n \to \infty} \sup \int\limits_{E} g_n \geq \lim\limits_{n \to \infty} \int\limits_{E} f_n$$
Also
$$\int\limits_{E} \lim\limits_{n \to \infty} \sup f_n \geq \lim\limits_{n \to \infty} \int\limits_{E} f_n \geq \int\limits_{E} \lim\limits_{n \to \infty} \inf f_n \ \   (**)$$
From (*) and (**) we have:
$$\int\limits_{E} \lim\limits_{n \to \infty} inf f_n \leq \lim\limits_{n \to \infty} inf \int\limits_{E} f_n \leq \lim\limits_{n \to \infty} \sup \int\limits_{E} f_n \leq \int\limits_{E} \lim\limits_{n \to \infty} \sup f_n$$ 
A: Hint: Apply Fatou's Lemma to the sequences $g + f_n, g- f_n.$
A: We can assume that $|f_n| \leq g$.
let $g_n= $inf $f_k < f_n $. $g_n \rightarrow$ lim inf $f_n$. Since $-g < f_n < g$ for all $n$, it follows that $-g < g_n < g$ thus $|g_n| \leq g$. Now by the lebesgue dominated convergence theorem we get $\int_{E}$ lim inf $f_n =$ lim $\int g_n =$ lim inf $\int_{E} g_n < $ lim inf $\int_{E} f_n$. by the monotonicity of the integral thus we have $\int_{E}$ lim inf $f_n \leq $ lim inf $\int_E f_n \leq$ lim sup $\int_E f_n$
