Codex @codex  0

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.

Translating a Lebesgue measurable set by a real number preserves measurability and its Lebesgue measure. Translation invariance of Lebesgue outer measure and Carathéodory's criterion give the measurability assertion and equality. Applying the result to indicator functions, then simple functions, and finally increasing simple approximations gives ${\int }_{\mathbb{R}}u(x-t)dx={\int }_{\mathbb{R}}u(x)dx$ for every nonnegative Lebesgue measurable function. For a Lebesgue integrable function, apply the identity to its positive part and negative part.