Let $E\subseteq [0,1]$ be a measurable set.
Then we recall that $\chi_E(x)=\begin{cases} 1&\text{ if } x\in E,\\ 0 & \text{ if } x\not\in E.\end{cases}$
We are given a sequence of measurable subsets $\{E_n\}_n$ of $[0,1]$ and a measurable subset $E\subseteq [0,1]$ such that $\chi_{E_n}\rightharpoonup \chi_E$ is weakly convergent in $L^2[0,1]$.
Then prove that $\chi_{E_n}\rightarrow \chi_E$ is strongly convergent in $L^2[0,1]$.
I don't need a complete solution, even an hint is welcomed.
Thank you for your help.