$\int f_k\to 0 $ but $f_k $ does not converge to $0 $ ae, where $ f_k $ is defined in $[0, 1] $ Give an exemple, in [0, 1], of a sequence of functions $ f_k $ such that $||f_k||_ 1=\int |f|_k \to 0 $ but $ f_k $ does not converge to $0 $ a.e.

Given a set $A$ recall that $\mathcal{X}_A$ is the characteristic function of $A$. 
Let $f_1=\mathcal{X}_{[0, 1]}$
$f_2=\mathcal{X}_{\left[0, \frac{1}{2}\right]}$
$f_3=\mathcal{X}_{\left[\frac{1}{2}, 1\right]} $
$f_4=\mathcal{X}_{\left[0, \frac{1}{3}\right]} $
$f_5=\mathcal{X}_{\left[\frac{1}{3}, \frac{2}{3}\right]} $
$f_6=\mathcal{X}_{\left[\frac{2}{3}, 1\right]} $
$f_7=\mathcal{X}_{\left[0, \frac{1}{4}\right]} $
and so on...
