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On $[0,1]$: $f_n = a_n\chi_{[\alpha n,\alpha n + \varepsilon n^{-2}]\ {\rm mod}\ 1 }$ with $\alpha$ irrational, and $a_n = 1$ or $a_n = n^2$.

This is, of course, also similar ...

Transferred from my comments below, and corrected (TeX was not shown, so I did not see that some of the code did not work):

Well, it was rather late (in my timezone). So I only typed in the $f_n$.

$\int |f_n -0|$ is either $= \varepsilon n^{-2} \to 0$ or $= \varepsilon \to \varepsilon$, respectively.

$\int \bigcup_{n\ge N} \lbrace x | \chi_{[\alpha n,\alpha n + \varepsilon n^{-2}]\ {\rm mod}\ 1 } \ne 0 \rbrace \le \sum_{n\ge N}\varepsilon n^{-2} \to 0$, i.e., $f_n \to 0$ almost everywhere.

On $[0,1]$: $f_n = a_n\chi_{[\alpha n,\alpha n + \varepsilon n^{-2}]\ {\rm mod}\ 1 }$ with $\alpha$ irrational, and $a_n = 1$ or $a_n = n^2$.