Fatou's Lemma for Integrals/Integrable Functions

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\left({f_n}\right)_{n \in \N} \in \mathcal{L}^1$, $f_n: X \to \R$ be a sequence of integrable functions.

Let $\displaystyle \liminf_{n \to \infty} f_n: X \to \overline{\R}$ be the pointwise limit inferior of the $f_n$.

Suppose that there exists an integrable $f: X \to \R$ such that for all $n \in \N$, $f \le f_n$ pointwise.

Then:


 * $\displaystyle \int \liminf_{n \to \infty} f_n \, \mathrm d\mu \le \liminf_{n \to \infty} \int f_n \, \mathrm d\mu$

where:


 * the integral sign denotes $\mu$-integration; and
 * the right-hand side limit inferior is taken in the extended real numbers $\overline{\R}$.