If $E\subset X$ is a null set, then $\mu^*(A) = \mu^*(A \cup E) = \mu^*(A\setminus E), \forall A \subset X$.
I felt tempted to assume that since A and E are disjoint, I could use the additive property, namely $\mu^*(A \cup E)= \mu^*(A)+\mu^*(E)= \mu^*(A)$. Am I doing it right?
Edit: clarify that $\mu^*$ is the outer measure.