whether we can put the limit inside the $L^\infty$ norm

Question

Assume $f\in L^\infty(\Omega)$, $f_i\to f\ \mathrm{a.e.}$, $f_i$ can be dominated by a $L^\infty$ function. My question is whether we can get $$\lim_{i\to\infty} \|f_i\|_{L^\infty(\Omega)}=\|f\|_{L^\infty(\Omega)}$$ or not.

Answer 1

Not in general. For example, take $f_n=\boldsymbol 1_{[0,1/n]}(x)$ for all $n$. You have that $|f_n|\leq 1\in L^\infty $, $f_n\to 0$ a.e., $\|f_n\|_{L^\infty }=1$ for all $n$, but $\|f\|_{L^\infty }=0$.