# Fatou's Lemma for Integrals

This proof is about Fatou's Lemma in the context of Integral of Positive Measurable Function. For other uses, see Fatou's Lemma.

## Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

### Positive Measurable Functions

Let $\sequence {f_n}_{n \mathop \in \N} \in \MM_{\overline \R}^+$, $f_n: X \to \overline \R$ 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:

$\displaystyle \int \liminf_{n \mathop \to \infty} f_n \rd \mu \le \liminf_{n \mathop \to \infty} \int f_n \rd \mu$

where:

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

### Integrable Functions

Let $\sequence {f_n}_{n mathop \in \N} \in \LL^1$, $f_n: X \to \R$ be a sequence of $\mu$-integrable functions.

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

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

Then:

$\displaystyle \int \liminf_{n \mathop \to \infty} f_n \rd \mu \le \liminf_{n \mathop \to \infty} \int f_n \rd \mu$

where:

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

## Source of Name

This entry was named for Pierre Joseph Louis Fatou.