I feel like the answer is right in front of me for this one, but for some reason I can't quite figure it out.
The questions ask us to prove that the Lebesgue extension of any measure $m$ is complete.
We are given the definition: A measure $\mu$ is said to be complete if every subset of a set of measure zero is measurable (i.e. if $A'\subset A$, $\mu(A)=0$ implies $A'\in \varphi_\mu$,) where $\varphi_\mu$ is the system of all measurable sets with $\sigma$-additive measure $\mu$.
I also know that If $A'\subset A$ and $\mu(A)=0$, then $\mu^*(A')=0$. And since the empty set, $\emptyset \in R(\varphi_m)$ then $\mu^*(A'\bigtriangleup \emptyset) = \mu^*(A')=0$.