Reverse Fatou's Lemma

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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.

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

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

where $\le$ signifies a pointwise inequality.


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


Then:

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

where:

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

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


Then:

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

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.


Also see