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Fatou's Lemma states that for any sequence of non-negative measurable functions $f_n$, the following inequality holds:

The Dominated Convergence Theorem can follow from Fatou's Lemma. If $f_n$ converges to $f$ a.e. and is dominated by an integrable function $g$, then Fatou's Lemma helps to show that

Fatou's Lemma is integral within the Lebesgue integration framework as it validates that integration is well-defined in the limit under certain conditions. As an example, consider a sequence of simple functions that approximate a non-negative measurable function; Fatou's Lemma assures that we can interchange limit and integration for this sequence.

Consider the sequence of functions $f_n(x) = (-1)^n \frac{1}{n} \chi_{[0,1]}(x)$, where $\chi$ is the characteristic function. Here, $\liminf_{n\to\infty} f_n(x) = 0$, but

Fatou's Lemma requires the functions in the sequence to be non-negative because the lemma relies on the Monotone Convergence Theorem, which in turn is only valid for non-negative functions. If we considered a sequence with negative parts, the lemma might fail as integrals might not be well-defined (e.g., result in $\infty - \infty$).