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Suppose that $A,B$ are separated sets of real numbers, that is $$\inf \{|a-b|:a \in A,b \in B\}>0.$$ Is it then true that $$m^*(A \cup B)=m^*(A)+m^*(B),$$ where $m^*$ is the Lebesgue outer measure? Does this relation hold for all outer measures?

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    it is not true with outer measure,but it is true with the Lebesgue measure.2012-11-05

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