$||f-f_n||_{L^1} \rightarrow 0$ but $f_n \rightarrow f$ for no $x$

Question

Show that there are $f\in L^1(\mathbb{R}^d)$ and a sequence $\{ {f_n}\}$ with ${f_n}\in L^1(\mathbb{R}^d)$ such that $||f-f_n||_{L^1} \rightarrow 0$ but $f_n \rightarrow f$ for no $x$

Thanks.

Answer 1

Hint: Think of a sequence of cubes whose volumes go to $0$ but cover each point of ${\mathbb R}^d$ infinitely often.

Answer 2

Consider the sets \begin{array}{llllllll} S_{1,1}:=[0,1]& & & & & & &\\ S_{2,1}:=[0,2^{-1})& S_{2,2}:=[2^{-1},1)& & & &&&\\ S_{3,1}:=[0,2^{-2})& S_{3,2}:=[2^{-2},2\cdot 2^{-2})&S_{3,3}:=[2\cdot 2^{-2},3\cdot 2^{-2})&S_{4,3}:=[3\cdot 2^{-2},1), \end{array} and more generally, $S_{n,k}:=[(k-1)2^{-n},k2^{-n})$. We can write it in lines, and $f_N$ is the characteristic function of these sets, going from the top to the bottom and left to right.