Prove that there must exist a measurable function $f$ on E and {$f_n$} converges to $f$ almost everywhere.

Question

Suppose $(X,R,\mu)$ is a measure space,{$f_n$} is a sequence of measurable function on E.If {$f_n$} is convergence almost everywhere on E. Prove that there must exist a measurable function $f$ on E and {$f_n$} converges to $f$ almost everywhere.
If $(X,R,\mu)$ is a complete measure space, I can find a measurable set $E_1$,where $\mu(E-E_1)=0$ and {$f_n$} convergent to a measurable function $f_1$ on $E_1$.We set $$f = \begin{cases} f_1, & \text{if $x\in E_1$} \\ 0, & \text{if $x\in E-E_1$ } \end{cases}$$ So $f$ is the measurable function on E.However,if $(X,R,\mu)$ is only a measurable space,how to prove $E_1$ is a measurable set.