Fatou's Lemma for Integrals

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This proof is about Fatou's Lemma for integrals. For other uses, see Fatou's Lemma.

Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.


Positive Measurable Functions

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:

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

where:

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


Integrable Functions

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 $\overline{\R}$.


Source of Name

This entry was named for Pierre Joseph Louis Fatou.


Also see