Let $(\Omega, \mathcal{F},P)$ be a space of probability. Let $f$ be a measurable function over $\mathcal{F}$, and $Z$ a subset of a set with measure zero. Then, define: \begin{equation*} \overline f =\left\{\begin{array}{lcr} f \,over\, Z^c\\ k \,over\, Z \end{array} \right. \end{equation*}
with $k$ a constant. Show that if $(\Omega, \mathcal{F},P)$ is complet, then $\overline f$ is measurable. Show that it can fail if $(\Omega, \mathcal{F},P)$ is not complet.
I know that if $(\Omega, \mathcal{F},P)$ is not complet, then $Z$ don't necessarily need to be in $\mathcal{F}$ and so $Z^c$. Is that enough? How do I put it formally? Is there an example for $(\Omega, \mathcal{F},P)$ not being complet and then $f$ fails to be measurable? I appreciate any help.
If $(\Omega, \mathcal{F}, P)$ is complete then $Z \in \mathcal{F}$ so $1_Z$ and $1_{Z^c}$ are measurable. In particular, $\bar{f} = f 1_{Z^c} + k 1_{Z}$ is measurable since products and sums of measurable functions are measurable.
If $(\Omega, \mathcal{F}, P)$ isn't complete then there is a null set $Z$ which is not measurable. In particular, if $f$ is the constant function $1$ and $k = 0$ then $\bar{f} = 1_{Z^c}$ is not measurable since $1_A$ is measurable iff $A$ is measurable and $Z^c$ is not measurable as the complement of a measurable set.