Meaure theory problem no. 12 page 92, book by stein and shakarchi Show that there are $f \in L^1(\Bbb{R}^d,m)$ and a sequence $\{f_n\}$ with $f_n \in L^1(\Bbb{R}^d,m)$ such that $\|f_n - f\|_{L^1} \to 0$, but $f_n(x) \to f(x)$ for no $x$.

Let us see the idea for  $\mathbb{R}$ . Let $f_1$ be a characteristic function of $[0,0.5]$, $f_2$ --- of [0.5,1], $f_3$ of [0,0.25] and so on. They form the desired sequence on $L^1([0,1])$. The rest is an exercise  because $\mathbb{R}$  can be covered by a countable number of intervals.
