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Prove the following generalization of Fatou's Lemma: if $\{f_n\}$ is a sequence of nonnegative measurable functions on $E$, then $\int_E \lim \inf f_n \le \lim \inf \int_E f_n$.

I found this, but I don't understand how or why the Monotone Convergence Theorem is of any help. Here's my proof with just relies on Fatou's Lemma.

By definition, $\inf_{k \ge n} f_k$ converges pointwise to $\lim \inf f_n$, so by Fatou's lemma we get

$$\int_E \lim \inf f_n \le \lim \inf \int_E \inf_{k \ge n} f_k$$ But $\inf_{k \ge n} f_k \le f_n$ for every $n \in \Bbb{N}$ which implies $\int_E \inf_{k \ge n} f_k \le \int_E f_n$ for every $n \in \Bbb{N}$ and finally

$$\lim \inf \int_E \inf_{k \ge n} f_k \le \lim \inf \int_E f_n$$ The above two inequalities yield

$$\int_E \lim \inf f_n \le \lim \inf \int_E f_n,$$ which concludes the proof.

Have I made a mistake that is presently eluding me?

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  • $\begingroup$ In fact DCT is not necessary for this version of Fatou's lemma (and some authors acrually call this one "Fatou's lemma"). I think I recall hearing during a lecture that Fatou (this one), Beppo Levi and DCT are three theorems that each author decides to prove in whatever order he prefers (don't quote me on that). For instance, Royden proves this Fatou and then Beppo Levi and DCT. $\endgroup$ – user228113 Mar 22 '18 at 18:07
  • $\begingroup$ @G.Sassatelli So the way I've proven it is okay? $\endgroup$ – user193319 Mar 22 '18 at 19:08
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    $\begingroup$ Yes, the proof is okay. This generalization is usually called 'Fatou's Lemma', see e.g. Wikipedia. $\endgroup$ – p4sch Mar 24 '18 at 8:39

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