Reverse Fatou's Lemma

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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.

Suppose that there exists a positive measurable function $f: X \to \overline{\R}$ such that:

$\displaystyle \int f \, \mathrm d\mu < +\infty$
$\forall n \in \N: f_n \le f$

where $\le$ signifies a pointwise inequality.


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


Then:

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

where:

the integral sign denotes $\mu$-integration; and
the left-hand side limit superior 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 $\displaystyle \limsup_{n \to \infty} f_n: X \to \overline{\R}$ be the pointwise limit superior of the $f_n$.

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


Then:

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

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