Can You Prove the Limit of Integrals Over Vanishing Measure Sets Is Zero?

[Euge]

Here is this week's POTW:

-----
Let $(X,\mathcal{M}, \mu)$ be a positive measure space, and let $\{E_n\}$ be a sequence of sets in $\mathcal{M}$ such that $\displaystyle\lim_n \mu(E_n) = 0$. Prove that if $1 \le p \le \infty$, then for all $f\in \mathscr{L}^p(X,\mathcal{M},\mu)$, $\displaystyle\lim_n \int_{E_n} f\, d\mu = 0$.-----

Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[Euge]

No one answered this week's problem. You can read my solution below.

Spoiler

If $q$ is the exponent conjugate to $p$, Hölder's inequality gives $\left\lvert\int_{E_n} f\, d\mu\right\rvert \le \|1_{E_n}\|_{\mathscr{L^q}} \|f\|_{\mathscr{L^p}} = \mu(E_n)^q\, \|f\|_{\mathscr{L^q}} \to 0$ as $n \to \infty$.