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 $\{N \in \mathscr M \mid \bar{\mu}(E) =0\}$.