Let $E \in \mathcal{L}$ and let $\mathbb{I}_E$ be the indicator function for $E$ in $\mathbb{R}^n$. Results
imply
\begin{equation}
\begin{split}
\int_{\mathbb{R}^n} \mathbb{I}_E(-x) \, d \mu(x) & = \int_{\mathbb{R}^n} \mathbb{I}_{- E}(x) \, d \mu(x) \\
& = \mu(-E) = \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]]]. $\square$
