For a set A is it always possible to find a measurable superset A* such that $\mu^*(A^*-A)=0$ Given a measure set $\langle X,\mu,\mathcal{A}\rangle$ let $\mu^*$ be the outer measure induced by $\mu$  and $A \subseteq X$ with finite outer measure (Not necessarily measurable). Can a measurable superset $A^*$ with the same outer measure as $A$ be found and has the property that $\mu^*(A^*-A)=0$? How do you prove you can find such a set? (Note: a set is a superset of itself I am not considering only strict supersets)
Edit: I have worked on this for a while and I believe it can be proved that this is never the case for non measurable sets for complete measures. If the outer measure of a set is $0$ then it is measurable according to Carathèodory. A non measurable set $A$ union a null set $N$ must be not measurable in a complete measure. This is because if $A\cup N$ was measurable then we would have that $A-N$ would have to be measurable and by completeness $A \cap N$ would also be measurable so $A$ would have to be measurable contradicting the premises.  Is this correct?

No, it’s not always possible. If $X=\{0,1\}$ with $\mu(A)$ the cardinality of $A$, then there is no such superset of any subset of $X$.
