If $f_n \to f$ in $L^2$, does $\chi_{\{f_n=0\}} \to \chi_{\{f=0\}}$ in $L^2$?

Question

Suppose we are on a bounded domain.

If $f_n \to f$ in $L^2$, does $\chi_{\{f_n=0\}} \to \chi_{\{f=0\}}$ in $L^2$? Here $\chi_A$ is the characteristic function of the set $A$. Could it hold for a subsequence if not the whole sequence?

Answer 1

If the domain is the unit interval and $f_n=1/n$, then $f_n\to 0$ in $L^2$ but $\chi_{\{f_n=0\}}=0$ and $\chi_{\{f=0\}}=1$.