Given $\{E_n\}$ be a sequence of measurable subsets of $[0,1]$ such that $$lim_{n,k\rightarrow\infty} \int\mid \chi_{E_n}-\chi_{E_k} \mid dm=0.$$
Prove that there is a measurable subset $E$ of $[0,1]$ such that $$lim_{n\rightarrow\infty} \int\mid \chi_{E_n}-\chi_E \mid dm=0.$$
This problem seems easy, but I haven't figure it out yet.
If $E =\lim \inf_n E_n$ then $\int |\chi_{E_n} -\chi_E|dm\leq \lim \inf_k \int |\chi_{E_n} -\chi_{E_k}|dm $ by Fatou's Lemma and this last quantity tends to $0$ as $n \to \infty$.