E is measurable, and E is a disjoint union of E1, E2. I want to show that if m(E)= m*(E1)+m*(E2) then E1 and E2 are measurable. How can I do this?
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5Is this your problem? – mishmish-dev Mar 21 '16 at 18:19
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If $m(E)=+\infty$, the result is false. Example: Take $E_1$ to be any non-measurable set in $\mathbb{R}$ and $E_2$ to be $\mathbb{R}\setminus E_1$. We have that $\mathbb{R}$ is the disjoint union of $E_1$ and $E_2$ and $$ +\infty =m(\mathbb{R})=m^(E_1)+m^(E_2)$$ However, $E_1$ and $E_2$ are not measurable. – Ramiro Mar 23 '16 at 14:12
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What if the condition added? E has finite measure. – user88914 Mar 23 '16 at 14:34