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Let $(X,\mathcal A, \mu)$ be a measure space and let $f\in L^p(\mu)$ for $1\leq p<\infty$. Let $E_n=\{x\in X: \frac{1}{n}\leq |f(x)|\leq n\}$ for all $n\in \mathbb N$. I have to show that $\mu(E_n)<\infty$. Can you please give a hint?

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    $E_n$ defined like this has no connection with $f$. You probably mean something like $\frac1{n}\leq f(x)\leq n$ – 2017-02-21
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    yes, thanks, I have edited accordingly – 2017-02-21

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Hint:

$\int|f|d\mu\geq\int_{E_n}|f|d\mu\geq\frac1{n}\mu(E_n)$

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    Then for all $n\in \mathbb N$, $\infty>\int |f|^pd\mu\geq \frac{1}{n}\mu(E_n)$. Thus $\mu(E_n)<\infty$. But where we have used $|f(x)|\leq n$? @ drhab – 2017-02-21
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    We just didn't need that. If $F_n:=\{x\in X:\frac1{n}\leq|f(x)|\}$ then it can be shown also that $\mu(F_n)<\infty$ (and consequently $\mu(E_n)<\infty$ since $E_n\subseteq F_n$). – 2017-02-21