$ ( X, \mu) $ is a complete measure space and $E_{n}$ are measurable sets such that $ \mu (E_{n}) < \infty$ for all $n$. Let $ \chi_{E_{n}} $ converge to $f$ in $ L_{1}$. Prove that $f$ is the characteristic function of a measurable set. (almost everywhere)
So far, I proved that $f = lim \chi _{E_{n_{k}}}$ for some sub sequence $E_{n_{k}}$ of $E_{n}$. But I don't know where to go from there. Is the limit of characteristic functions a characteristic function? because i was not able to prove that.
thank you