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{{theorem|monotone convergence theorem|Let <math>(\Omega, \mathcal F, \mu)</math> be a measure space, and let <math>f_n: \Omega \to [0, \infty]</math> be an ascending (that is, <math>f_{n+1} \ge f_n</math> pointwise) sequence of non-negative functions, that converges pointwise to a function <math>f: \Omega \to [0, \infty]</math>. Then
:<math>\lim_{n \to \infty} \int_\Omega f_n d\mu = \int_\Omega f d\mu</math>.}}

{{theorem|Fatou's lemma|Let <math>(\Omega, \mathcal F, \mu)</math> be a measure space, and let <math>f_n: \Omega \to \mathbb R_{\ge 0}</math> be a sequence of non-negative functions. Then
:<math>\int_\Omega \liminf_{n \to \infty} f_n d\mu \le \liminf_{n \to \infty} \int_\Omega f_n d\mu</math>.}}

{{proof|Note that, upon defining
:<math>g_N := \inf_{k \ge N} f_n</math>,
that the sequence of functions <math>g_N</math> is strictly ascending and converges pointwise to <math>f</math> as <math>N \to \infty</math>. Hence, the [[Measure Theory/Convergence theorems#monotone convergence theorem|monotone convergence theorem]] is applicable and we obtain
:<math>\lim_{N \to \infty} \int_\Omega g_N d\mu = \int_\Omega f d\mu</math>.
Now for each <math>N \in \mathbb N</math>, we have
:<math>\int_\Omega g_N d\mu \le \int_\Omega f_N d\mu</math>,
and if we take the lim inf,
:<math>\lim_{N \to \infty} \int_\Omega g_N d\mu = \liminf_{N \to \infty} \int_\Omega g_N d\mu \le \liminf_{N \to \infty} \int_\Omega f_N d\mu</math>.}}

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