Let $y \in \mathbb{R}^n$. Let $E \in \mathcal{L}$ and let $\mathbb{I}_E$ be the indicator function for $E$ in $\mathbb{R}^n$. Results
then imply
\begin{equation}
\begin{split}
\int_{\mathbb{R}^n} \mathbb{I}_E(x + y) \, d \mu(x) & = \int_{\mathbb{R}^n} \mathbb{I}_{E - y}(x) \, d \mu(x) \\
& = \mu(E - y) = \mu(E) = \int_{\mathbb{R}^n} \mathbb{I}_E(x) \, d \mu(x)
\end{split}
\end{equation}
This establishes the claim for measurable indicator functions $\mathbb{R}^n \to \{ 0, 1 \}$. The claim for unsigned functions $\mathbb{R}^n \to [0, \infty]$ then follows by applying the principles in [[[x,125]]]. $\square$