MHB Outer measure exclusion of zero set question

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I've just started self-studying measure theory by reading Pugh's Mathematical Analysis. I'm trying to understand his argument for why the exclusion of a zero set does not change the outer measure: $m^*(E\setminus Z)=m^*(E)$:

(Pugh's arugment): 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 $m^*(E\setminus Z)=m^*((E\setminus Z)\cup(E\cap Z))=m^*(E).$ QED

My question is where does the $E\cap Z$ come from in the second equality above? If you're using $m^*(E)=m^*(E\cup Z)$ and making the substitution $E\to E\setminus Z$, why isn't it $m^*(E\setminus Z)=m^*((E\setminus Z)\cup Z)?$

Also, this is my first post on this site, so I apologize if something isn't formatted correctly.
 
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Adam1729 said:
I've just started self-studying measure theory by reading Pugh's Mathematical Analysis. I'm trying to understand his argument for why the exclusion of a zero set does not change the outer measure: $m^*(E\setminus Z)=m^*(E)$:

(Pugh's arugment): 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 $m^*(E\setminus Z)=m^*((E\setminus Z)\cup(E\cap Z))=m^*(E).$ QED

My question is where does the $E\cap Z$ come from in the second equality above? If you're using $m^*(E)=m^*(E\cup Z)$ and making the substitution $E\to E\setminus Z$, why isn't it $m^*(E\setminus Z)=m^*((E\setminus Z)\cup Z)?$

Also, this is my first post on this site, so I apologize if something isn't formatted correctly.
I think the following is being said: Since $Z$ is a zero set, so is $E\cap Z$. Thus $m^*(A)=m^*(A\cup (E\cap Z))$ for any $A\subseteq \mathbb R$. In particular we can take, $A=E\setminus Z$.

Does this answer your question?
 
Yes, thank you very much! :)
 
I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...
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