= Translation invariance of Lebesgue measure
{title2=$m(E+t)=m(E)$}
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>.
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