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?
If $A$ and $B$ are separated, is $m^*(A \cup B)=m^*(A)+m^*(B)$?
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measure-theory
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0it is not true with outer measure,but it is true with the Lebesgue measure. – 2012-11-05