Let \(f\colon\RR^d\to[0,\infty]\) be Lebesgue measurable, or let \(f\in L^1(\RR^d)\), and let \(h\in\RR^d\) and \(\lambda\in\RR\setminus\theset{0}\). Then \(x\mapsto f(x+h)\) and \(x\mapsto f(\lambda x)\) are measurable, and \[ \int_{\RR^d} f(x+h)\dx = \int_{\RR^d} f(x)\dx, \qquad \int_{\RR^d} f(\lambda x)\dx = \abs{\lambda}^{-d}\int_{\RR^d} f(x)\dx. \]
We use that for a Lebesgue measurable \(E \subseteq \RR^d\) the sets \(E + h \coloneqq \theset{x + h \suchthat x \in E}\) and \(\lambda E\coloneqq\theset{\lambda x\suchthat x\in E}\) are measurable, with \(m(E + h) = m(E)\) and \(m(\lambda E) = \abs{\lambda}^d m(E)\). Put \(\tau f(x)\coloneqq f(x+h)\) and \(\delta f(x)\coloneqq f(\lambda x)\). For a Borel set \(B\), \(\inverseof{(\tau f)}(B) = \inverseof{f}(B) - h\) and \(\inverseof{(\delta f)}(B) = \lambda^{-1}\inverseof{f}(B)\), so \(\tau f\) and \(\delta f\) are measurable.
Characteristic functions. For measurable \(E\), \(\tau\chi_E = \chi_{E-h}\) and \(\delta\chi_E = \chi_{\lambda^{-1}E}\), so \[ \int\tau\chi_E = m(E - h) = m(E) = \int\chi_E, \qquad \int\delta\chi_E = m(\lambda^{-1}E) = \abs{\lambda}^{-d}\int\chi_E. \]
Simple functions. A nonnegative simple function is a finite sum \(\phi = \sum_i c_i \chi_{E_i}\) with \(c_i\geq 0\) and \(E_i\) measurable, and \(\tau\phi = \sum_i c_i\tau\chi_{E_i}\) and \(\delta\phi = \sum_i c_i\delta\chi_{E_i}\). By the case of characteristic functions, \(\int\tau\phi = \int\phi\) and \(\int\delta\phi = \abs{\lambda}^{-d}\int\phi\).
Nonnegative measurable functions. By the definition of the integral, \(\int f\) is the supremum of \(\int\phi\) over simple \(0 \leq \phi \leq f\). The maps \(\phi\mapsto\tau\phi\) and \(\phi\mapsto\delta\phi\) are bijections from the simple functions \(0\leq\phi\leq f\) onto the simple functions \(0\leq\psi\leq\tau f\) and \(0\leq\psi\leq\delta f\) respectively, with inverses given by translation by \(-h\) and dilation by \(\lambda^{-1}\). They multiply integrals by \(1\) and by \(\abs{\lambda}^{-d}\), so the suprema satisfy \(\int\tau f = \int f\) and \(\int\delta f = \abs{\lambda}^{-d}\int f\).
Integrable functions. For real-valued \(f\in L^1(\RR^d)\), write \(f = f^+ - f^-\) with \(f^\pm \geq 0\); then \(\tau f = \tau f^+ - \tau f^-\) and \(\delta f = \delta f^+ - \delta f^-\), and the nonnegative case applies to each part. For complex-valued \(f\), apply the real case to \(\operatorname{Re} f\) and \(\operatorname{Im} f\).