TITLE: $||f-f_n||_{L^1} \rightarrow 0$ but $f_n \rightarrow f$ for no $x$
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: Hint: Think of a sequence of cubes whose volumes go to $0$ but cover each point of ${\mathbb R}^d$ infinitely often.