Convergence of $\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$ for $u_n \to u$?

Question

Let $u_n \to u$ in $L^2(\Omega)$ and let $I$ be an bounded interval.

Does it follow that $$\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$$ at least for a subsequence of $u_{n_j}$ (because we will have pointwise convergence for $u_{n_j} \to u$ a.e.)

It appears true but no idea how to prove it.

Answer 1

Consider the following counter example: $f_n \equiv 1 - 1/n$ on $\Omega = (0, 3)$ (you could also take any finite measure space), $I = [1, 4)$ and $f\equiv 1$. You can also take $I = (0,1)$ or $ I = [1,4]$, showing that this fails for all types of intervals.