 Solution

If the intervals $J_k$ cover $A$, their translates $J_k+a$ cover $A+a$ and have the same lengths. Every candidate total length in the definition of $m^{\ast }(A)$ is therefore also a candidate for $m^{\ast }(A+a)$. Taking infima gives $m^{\ast }(A+a)\le m^{\ast }(A)$. Apply the same argument to $A+a$ and the translation $-a$ to obtain the reverse inequality. HenceThis translation invariance of Lebesgue outer measure holds for every subset $A$, including non-measurable sets and sets of infinite outer measure.