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

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Chris L T521

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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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