How to show $m(A) = m(A \backslash Z)$

Question

Let $m$ be the Lebesgue measure, and $Z$ a set of measure zero, $A \subset \mathbb{R}$

Then intuitively, $m(A) = m(A \backslash Z)$

How to show this?

Attempt: $m(A \backslash Z) = m(A \cap Z^c)$ Is there a way to turn that $\cap$ upside down and use countable additivity?

$A-Z \subset A$, and so $m(A-Z) = m(A) - m(Z) = m(A)$. Or you can say that $A-Z$ and $Z$ are disjoint subsets which union to $A$, so $m(A-Z) + m(Z) = m(A)$ — Chris Rackauckas, Apr 24 '16 at 00:36

score 2 · Accepted Answer · answered Apr 24 '16 at 00:36

2

$m(A)=m(A \cap Z)+ m(A \cap Z^C)$

but we know that $0 \leq m(A \cap Z) \leq m(Z)=0$.

Hence $m(A)=m(A \cap Z^C).$

answered Apr 24 '16 at 00:36

Siong Thye Goh

Okay, and the first line holds because we already know that $Z$ is measurable, therefore the caratheodory condition holds $m^(A) = m^(A \cap Z) + m^(A \cap Z^c)$ and $m^$ coincides with $m$ right? – Fraïssé Apr 24 '16 at 00:51
yup. that is right. – Siong Thye Goh Apr 24 '16 at 01:45

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