Fatou's Lemma for Measures/Corollary

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Corollary to Fatou's Lemma for Measures

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

Let $\sequence {E_n}_{n \mathop \in \N} \in \Sigma$ be a sequence of $\Sigma$-measurable sets.

Let $\mu$ be a finite measure.


Then:

$\displaystyle \map \mu {\limsup_{n \mathop \to \infty} E_n} \le \limsup_{n \mathop \to \infty} \map \mu {E_n}$

where:

$\displaystyle \limsup_{n \mathop \to \infty} E_n$ is the limit superior of the $E_n$
the right hand side limit superior is taken in the extended real numbers $\overline \R$.

Proof


Examples

Union of Disjoint Real Intervals

Letting $E_n = \openint n {n + 1}$.

Since the $E_n$ are pairwise disjoint, the definition of limit superior gives:

$\displaystyle \limsup_{n \mathop \to \infty} E_n = \O$

By Measure of Interval is Length, we also have:

$\map \mu {E_n} = 1$

for all $n \in \N$.


Thus:

$0 = \displaystyle \map \mu {\limsup_{n \mathop \to \infty} E_n} < \limsup_{n \mathop \to \infty} \map \mu {E_n} = \limsup_{n \mathop \to \infty} 1 = 1$


Source of Name

This entry was named for Pierre Joseph Louis Fatou.


Sources