TITLE: $\int f_k\to 0 $ but $f_k $ does not converge to $0 $ ae, where $ f_k $ is defined in $[0, 1] $
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: Define $I_{n,k}:=[k2^{-n},(k+1)2^{-n})$ for $n\in\mathbb N$ and $0\leqslant k\leqslant 2^n-1$. Then define 
$$f_1=a_1\chi_{I_{1,0}}, f_2:=a_1\chi_{I_{1,1}}$$
$$f_3=a_2\chi_{I_{2,0}}, f_4=a_2\chi_{I_{2,1}}, f_5=a_2\chi_{I_{2,2}}, f_6=a_2\chi_{I_{2,3}}$$
$$\vdots$$
$$f_{2^{n+1}+1+k}:=a_n\chi_{I_{n,k}},\quad  0\leqslant k\leqslant 2^n-1.$$
Taking $a_n$ such that $a_n2^{-n}\to 0$, we have $\lVert f_n\rVert_{L^1}\to 0$, but there is not convergence almost everywhere to $0$ (because for each $x$, there is $A(x)\subset \mathbb N$ infinite such that $f_j(x)>1$ for all $j\in A(x)$. 
However, it is true that if $f_n\to 0$ in $L^1$, then there is $n_k\uparrow \infty$ such that $f_{n_k}\to 0$ almost everywhere: we consider $n_k$ such that $\mu\{|f_{n_k}|>2^{-k}\}\lt 2^{-k}$. It holds for any measure space, not only on the unit interval endowed with Lebesgue measure.