Translation invariance of Lebesgue outer measure
= Translation invariance of Lebesgue outer measure
{title2=$m^*(A+a)=m^*(A)$}
Translating every <interval> of a countable cover of $A\subseteq\mathbb R$ by $a$ produces a cover of $A+a$ with the same lengths. Taking the <infimum> of cover lengths gives $m^*(A+a)\leq m^*(A)$. Translating back by $-a$ gives the reverse inequality. This proves <translation invariance> for <Lebesgue outer measure> on every <subset>, without first assuming the set is <Lebesgue measurable>.