How to Prove the Inequality for a Limit of Functions and Integrals?

[Chris L T521]

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]

This week's problem was correctly answered by Ackbach. You can find his solution below.

Spoiler

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.