Elaboration of a technique used in the monotone convergence theorem proof.

Question

The theorem and its proof are given below:

My question is: I do not understand why $\int_{E} \leq \lim \inf \int_{E} f_{n}$ and $\lim \sup \int_{E} f_{n} \leq \int_{E} f $ leads to $\int_{E} f = \lim_{n \rightarrow \infty} \int f_{n}$? could anyone explains this for me, please?

Answer 1

$$\limsup_n\int_Ef_n \leq \int_Ef \leq \liminf_n \int_Ef_n$$

So since $\liminf_n \int_Ef_n \leq \limsup_n\int_Ef_n$ we have form the above inequality that $\liminf_n\int_Ef_n=\int_Ef$ and

$\limsup_n\int_Ef_n=\int_Ef$

Answer 2

Unlike $\lim$, $\limsup$ and $\liminf$ always exist and equal real numbers (or $+\infty$ or $-\infty$, respectively), and satisfy $\liminf\leq \limsup$. The special case where $\liminf=\limsup$ is equivalent to $\lim$ exists.

(I should point out that the above statement holds for any sequences and doesn't have anything special to do with integration)