Measurability of a function with respect to the completion of a measure space I would like some help with the following proof. Thanks for any help in advance. 

Let$(X,\mathscr M, \mu)$ be a measure space and $(X,\overline{\mathscr M}, \overline{\mu})$ its completion. Show, if $f:X\to \overline{\mathbb R}$ is a $\overline{\mathscr M}$-measurable function, then there is an $\mathscr M$-measurable function g such that $$\overline{\mu}\{x:f(x)\ne g(x)\}=0.$$

Edit: I have been given the following hint. I may wish to use the observation that,
$f:X\to \overline{\mathbb R}$ is measurable if and only if $\{x : f(x) > q\}$ is measurable for every $q \in \mathbb{Q}$.
Edit 2: As per Saz's request, here is a definition,
If $(X,\mathscr M, \mu)$ has the property that F ∈$\mathscr M$ whenever $E \in \mathscr M$, ${\mu}(E) = 0$, and $F \subset E$, then $\mu$ is complete.
Furthermore $\overline{\mathscr M}$ is defined to be the set $\{E\cup F \mid E \in \mathscr M, F \in N$ for some $N \in \mathscr N\}$ where $\mathscr N$ is defined to be the set $\{N \in \mathscr M \mid \bar{\mu}(E) =0\}$.
 A: Hints:


*

*Consider the particular case that $f = 1_A$ is the indicator function of a set $A \in \bar{\mathcal{M}}$. Show that there exists a set $B \in \mathcal{M}$ such that $\bar{\mu}(1_B \neq 1_A)=0$.

*Extend the result to simple functions.

*Let $f \geq 0$ be an $\bar{\mathcal{M}}$-measurable function. Then there exists a sequence of simple functions $(f_n)_{n \in \mathbb{N}}$ which are $\bar{\mathcal{M}}$-measurable such that $f_n \geq 0$ and $f_n \uparrow f$ (i.e. $f_1(x) \leq f_2(x) \leq \ldots$ and $f(x) = \sup_{n \in \mathbb{N}} f_n(x)$ for all $x \in X$). By step 2, there exists a sequence $(g_n)_{n \in \mathbb{N}}$ of simple functions which are $\mathcal{M}$-measurable and satisfy $\bar{\mu}(f_n \neq g_n)=0$ for all $n \in \mathbb{N}$. Show that   $$g(x) := \sup_{n \in \mathbb{N}} g_n(x), \qquad x \in X$$ is $\mathcal{M}$-measurable and that $\bar{\mu}(f \neq g)=0$.

*For an arbitrary $\bar{\mathcal{M}}$-measurable function $f$ write $f= f^+ - f^-$ and apply step 3 to the positive part $f^+$ and negative part $f^-$, respectively.

