$f$ is measurable in complete measure space, then there is a $g$ measurable in the uncomplete space such that $f = g$ almost everywhere? I am trying to understand the proof a theorem in Folland. The theorem says if $(X,M,\mu)$ is a measure space with completion $(X,\bar{M},\bar{\mu})$, and $f$ is $\bar{\mu}$-measurable, then there is a $m$-measurable $g$ such that $f=g$, $\bar{\mu}$-almost everywhere.
He says this is obvious if $f = \chi_E$, the characteristic/indicator function of set $E$. I am not really sure I understood why the theorem is true in this case.
Attempt: If $E$ is not a null set, then we can let $g = \chi_E$. If $E$ is a null set, then $E$ is either in $M$ or $E$ is a subset of a null set in $M$ by definition of completion. If $E$ is in $M$, then let $g = \chi_E$. If not, it is a subset of some null set $S$. Let $g = \chi_S$, so $f = g$ except on $S - E$, which is a null set. Also, we know from earlier in the book that characteristic functions are measurable.
Question: Is my attempt correct? Did I miss out or misunderstood anything?

If $E\in\overline{\mathcal M}$, then there exist $E_1,E_2\in {\mathcal M}$, such that 
$$
E_1\subset E\subset E,\,\,\mu(E_2\smallsetminus E_1)=0 \quad \text{and}\quad \overline{\mu}(E)=\mu(E_1)=\mu(E_2).
$$
Thus
$$
\chi_{E_1}=\chi_{E_2}=\chi_E\quad \text{$\overline{\mu}$-a.e.}
$$
For a general $f$, we assume that $f\ge 0$, as if not we can $f$ as $f=f_+-f_-$, where $f_+,f_-$ are its positive and negative parts. Then $f$ can be written as
$$
f=\sum_{n\in\mathbb N} a_n\chi_{A_n},
$$
where $a_n>0$ and the $A_n$'s are $\overline{\mu}$-measurable. Choose $B_n\in{\mathcal M}$, s. t. $\chi_{A_n}=\chi_{B_n}$, $\,\,\overline{\mu}$-a.e., and a desired $\mu$-measurable function,  $\overline{\mu}$-a.e. equal to $f$, would be
$$
g=\sum_{n\in\mathbb N} a_n\chi_{B_n}.
$$
