Fatou's Lemma for Measures/Corollary/Examples/Union of Disjoint Real Intervals

Example of Use of Corollary to Fatou's Lemma for Measures

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:

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