Fatou's Lemma for Measures/Corollary/Examples
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Examples of Use of Corollary to Fatou's Lemma for Measures
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:
- $0 = \ds \map \mu {\limsup_{n \mathop \to \infty} E_n} < \limsup_{n \mathop \to \infty} \map \mu {E_n} = \limsup_{n \mathop \to \infty} 1 = 1$