= 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 $m^*(A)$ is therefore also a candidate for $m^*(A+a)$. Taking <infima> gives $m^*(A+a)\leq m^*(A)$. Apply the same argument to $A+a$ and the translation $-a$ to obtain the reverse inequality. Hence
$$
\boxed{m^*(A+a)=m^*(A).}
$$
This <translation invariance of Lebesgue outer measure> holds for every <subset> $A$, including non-measurable sets and sets of infinite outer measure.
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