I've just started self-studying measure theory by reading Pugh's Mathematical Analysis. He shows that the exclusion of a zero set does not change the outer measure: $m^*(E\setminus Z)=m^*(E)$, but there is a small thing I'm not understanding in his argument:
Let $Z$ be a zero set, $E\subseteq\mathbb{R}$, and $m^*$ be the Lebesgue outer measure. Since $m^*(E)=m^*(E\cup Z)$, applying this to the set $E\setminus Z$ gives $m^*(E\setminus Z)=m^*((E\setminus Z)\cup(E\cap Z))=m^*(E).$
My question is where does the $E\cap Z$ come from in the second equality? If you're using $m^*(E)=m^*(E\cup Z)$ applied to $E\setminus Z$, why isn't it $m^*(E\setminus Z)=m^*((E\setminus Z)\cup Z)?$
$(E\setminus Z)\cup(E\cap Z)=E$, so the equation in the second paragraph is just the one in the first paragraph with $m^*(E)$ rewritten.