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Let $(S,A,u)$ be a measure space and $f_n:S \rightarrow \mathbb{R}$ $ \cup$ {$-\infty$,$\infty$}. There is a $g$, which is a non-negative integrable function in $S$ with $f_n≥-g$ for all $n$. Then: $$ \int_{S}\liminf_{n \rightarrow \infty} f_n du\leq\liminf_{n \rightarrow \infty} \int_{S}f_n du $$

I found this theorem in a book, but without proof (that's why I found no suitable name for my problem).
I thought about setting $g$ as the zero function. This should be correct, because of Fatou's lemma. But do I have to show it for an unknown function $g$? If so, how can I prove it?

2 Answers 2

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Apply Fatou to $g+f_n$, using the fact that $\liminf_n(g+f_n)=g+\liminf_nf_n$.

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    Ok, my last problem is that I have to show that $g$ ist measurable. Maybe I just don't see the simple solution. I think I tried most of the criterias. Can you give me last hint? – 2017-01-26
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This is a slightly more general statement than Fatou's Lemma. In Fatou's Lemma, $f_n$ is non-negative, whereas here it is bounded below by a function with certain properties. The proof is pretty similar to Fatou's Lemma, with some extra work where you usually use non-negativity.