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If $E,F$ are countable union of pairwise disjoint intervals ,prove that $m^*(E\cup F)+m^*(E\cap F)=m^*(E)+m^*(F)$.

$E=\cup I_n,F=\cup J_n;I_p\cap I_q=\emptyset;J_p\cap J_q=\emptyset;p\neq q$

Then $m^*(\cup I_n\cup \cup J_n)=m^*(\cup (I_n\cup J_n))\le\sum m^*(I_n\cup J_n)\le \sum (m^*(I_n)+m^*(J_n))=\sum m^*(I_n)+\sum m^*(J_n)$.

But I am not getting the desired equality. How should I approach the problem?

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    this is false if $E = F$. – 2017-01-02
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    I'm guessing that the pairwise disjointness applies through all the intervals in E and F combined? – 2017-01-02

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