$f_n \to f$ a.e and $\|f_n\|_\infty \to \|f\|_\infty$, counterexample/(proof?) to $\|f_n-f\|_\infty \to 0$

Question

Here: If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$ is a proof for $1 \leq p <\infty$. Is the statement true for $L^\infty$ in general? Perhaps only on finite measures?

My first instinct was (on finite measure) to somehow use egorovs theorem. Since convergence in $L^\infty$ is equivalent to convergence almost uniformly. However, we need the measure to be $0$ and not $\epsilon$ as it is in the case of Egorovs.

Answer 1

$f_{n}=\chi_{[0,1-1/n]}$, $f_{n}\rightarrow\chi_{[0,1]}$ a.e. $\|f_{n}\|_{L^{\infty}[0,1]}=\|f\|_{L^{\infty}[0,1]}=1$, but $f_{n}-\chi_{[0,1]}=-\chi_{(1-1/n,1]}$ and hence $\|f_{n}-\chi_{[0,1]}\|_{L^{\infty}[0,1]}=1$ which does not converge to $0$.