If $\displaystyle (f_n)$ is a sequence of characteristic function of sets in $\displaystyle X$ and if $\displaystyle (f_n)$ converges to $\displaystyle f$ in $\displaystyle L_p(X,M_x,m)$,show that $\displaystyle f$ is (almost everywhere equal to) the characteristic function of a set in $\displaystyle X$.

Since $\displaystyle (f_n)$ converges in $\displaystyle L_p$,then it is convergent in measure.I try to connect things in this way but I am stuck.Can anyone give me any hints to proceed?