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:
- $\ds \map \mu {\limsup_{n \mathop \to \infty} E_n} \le \limsup_{n \mathop \to \infty} \map \mu {E_n}$
where:
- $\ds \limsup_{n \mathop \to \infty} E_n$ is the limit superior of the $E_n$
- the right hand side limit superior is taken in the extended real numbers $\overline \R$.
Proof
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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:
- $\ds \limsup_{n \mathop \to \infty} E_n = \O$
By Measure of Interval is Length, we also have:
- $\map \mu {E_n} = 1$
for all $n \in \N$.
Thus:
- $0 = \ds \map \mu {\limsup_{n \mathop \to \infty} E_n} < \limsup_{n \mathop \to \infty} \map \mu {E_n} = \limsup_{n \mathop \to \infty} 1 = 1$
Source of Name
This entry was named for Pierre Joseph Louis Fatou.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 9$: Problem $9 \ \text{(iii)}$