Generalization of Fatou's Lemma

Fatou's Lemma: Let $\{f_n\} \rightarrow f$ pointwise a.e. on $E$, then $\int_E f \leq \liminf \int_E f_n$.

Generalization: Prove that if $\{ f_n \}$ is a sequence of nonnegative measurable functions on $E$, then $$\int_E \liminf f_n \leq \liminf \int_E f_n$$

This is from Royden 4e (pg 85).

The general form does not require pointwise convergence. Is this so simple as to observe that the $\inf$ of simple functions will be less than the integral of the same? Otherwise I need some guidance. When can I pass the limit through the integral.

Solution 1:

If you are allowed to use monotone convergence theorem then

Observe that if $g_{n} := \inf_{k\geq n}f_{k}$ then $\int g_{n} \leq \int f_{n}$ and $g_{n}$ increases to $\lim \inf f_{n}$.