Fatou's Lemma for Integrals/Positive Measurable Functions

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

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

Then:


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

where:


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

Proof
The sequence $\displaystyle \inf_{m \mathop > n} f_m$ is a monotone increasing sequence.

Also:
 * $\displaystyle \inf_{m \mathop > n} f_m \le f_k$

for $k>n$.

We have then:
 * $\displaystyle \int \inf_{m \mathop > n} f_m \, \mathrm d \mu \le \int f_k \, \mathrm d \mu$

for $k > n$.

Thus:
 * $\displaystyle \int \inf_{m \mathop > n} f_m \, \mathrm d \mu \le \inf_{m \mathop > n} \int f_m \, \mathrm d \mu$

Now we take the limit $n \to \infty$.

Using the monotone convergence theorem we exchange integral and limit on the left side of the equation and obtain:
 * $\displaystyle \int \liminf_{m \mathop > n} f_m \, \mathrm d \mu \le \liminf_{m \mathop > n} \int f_m \, \mathrm d \mu$

Also see

 * Reverse Fatou's Lemma/Positive Measurable Functions