Understanding what Im supposed to prove in the following Fatou´s generalized lemma.

Question

The following exercise is claimed to be a Fatou´s generalized lemma:

Let $f_{n}:X \to \mathbb{R}$ a sequence of measurable functions and let suppose the is an $g:X \to \mathbb{R}$ such $-g \leq f_{n}$ for every $n \in \mathbb{N}$, then

$$\int_{X} lim inf \:f_{n} d \mu \leq lim inf \int_{X} f d \mu.$$

First of all, I dont understand why this is a generalization over Fatou´s lemma. I guess this is because the inequality above is stated over $X$ and not some $E$ in the $\sigma$-algebra. Also dont get why putting an extra condition like $-g \leq f_{n}$ make the original statement of Fatous lemma a generalization. And also, and the most important question what Im supposed to prove in this exercise and any hint how to do this? Thanks! And sorry, but I dont know which book my teacher got this exercise so Im not sure if there is any mistake in the statement. Aprecciate any help in order to understand and solve this problem.

Answer 1

This requires some hypothesis on $g$. I believe $g$ is supposed to be integrable. In this case we get this from Fatou's Lemma as follows:

$f_n+g \geq 0$ so $\int \lim \inf (f_n+g) \leq \lim \inf \int (f_n+g)$. This is same as $\int \lim \inf f_n +\int g \leq \lim \inf \int f_n+\int g$. Just cancel $\int g$ to get the result.

Why is this a generalization of Fatou's Lemma just put $g=0$ and you get Fatou's Lemma.

Integrating over $X$ or integrating over some measurable subset makes no difference. You can always replace $f_n$ by $f_nI_E$.

Answer 2

Assume that $g$ is integrable:

Note that $f_{n}+g\geq 0$, we have by usual Fatou that \begin{align*} \int\liminf(f_{n}+g)\leq\liminf\int(f_{n}+g). \end{align*} But \begin{align*} \int\liminf(f_{n}+g)=\int(\liminf f_{n})+g=\int\liminf f_{n}+\int g \end{align*} and \begin{align*} \liminf\int(f_{n}+g)=\left(\liminf\int f_{n}\right)+\int g. \end{align*} Now canceling each side by the real number $\displaystyle\int g$.