Translation invariance of Lebesgue outer measure ($m^{\ast }(A+a)=m^{\ast }(A)$)

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 $m^{\ast }(A+a)\le m^{\ast }(A)$. 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.