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

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#### Chris L T521

##### Well-known member
Staff member
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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#### Chris L T521

##### Well-known member
Staff member
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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