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Problem of the Week #60 - July 22nd, 2013

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Chris L T521

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Jan 26, 2012
995
Here's this week's problem.

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Problem: Consider\[f_n = \begin{cases}1 & \forall\,x\in\left[n,n+1\right)\\ 0 & \forall\,x\in\mathbb{R}\backslash\left[n,n+1\right)\end{cases}\]
Show that
\[\int_{\mathbb{R}}\liminf_{n\to\infty}f_n\,dm < \liminf_{n\to\infty}\int_{\mathbb{R}}f_n\,dm.\]

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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Chris L T521

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Jan 26, 2012
995
This week's problem was correctly answered by Ackbach. You can find his solution below.

This follows directly from Fatou's Lemma, but you can also simply compute both sides directly:

Note that
$$ \liminf_{n \to \infty} \int_{ \mathbb{R}}f_{n} \, dm= \liminf_{n \to \infty} \left[1 \cdot m([n,n+1)) + 0 \cdot m( \mathbb{R} \setminus [n,n+1)) \right]
= \liminf_{n \to \infty}1=1.$$
But
$$ \int_{ \mathbb{R}} \left[ \liminf_{n \to \infty} f_{n} \right] \, dm= \int_{ \mathbb{R}} 0 \, dm = 0.$$

Since $0<1$, we are done.
 
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