Exist measurable functions $f_n$ with $\limsup_{n\to\infty} f_n (x) = \infty$ but $\lim_{n\to\infty} \int f_n = 0$?

Question

Are there measurable functions $f_n: [0,1] \to [0, \infty)$, $n \in \mathbb{N}$, with $$\limsup_{n\to\infty} f_n (x) = \infty \qquad \forall x \in [0,1],$$ but $$\lim_{n\to\infty} \int_{[0,1]} f_n (x) dx= 0\ ?$$

Answer 1

You can use a variant of the typewriter sequence. Let $f_1(x)=\chi_{[0,1]}$, $f_2(x)=2\chi_{[0,1/2]}$, $f_3(x)=2\chi_{[1/2,1]}$. Next $f_4$, $f_5$, $f_6$ and $f_7$ are $3$ times the characteristic functions of the intervals $[0,1/4]$, $(1/4,2/4]$, $(2/4,3/4]$ and $(3/4,1]$. I hope the pattern is clear.