Let $M = (\mathbb{R}, \mathcal{L}, \ell)$ be a D1744: Lebesgue measure space such that
| (i) | $f : \mathbb{R} \to [0, \infty]$ is a D5610: Unsigned basic Borel function on $M$ |
| (ii) | $T : \mathbb{R} \to \mathbb{R}$ is a D3207: Linear function from $\mathbb{R}$ to $\mathbb{R}$ |
| (iii) | $T$ is an D3393: Invertible function |
| (iv) | $T^{-1}$ is an D4024: Inverse function for $T$ |
Then
| (1) | \begin{equation} \int_{\mathbb{R}} \left( f \circ T^{-1} \right) d \ell = |\text{det} T| \int_{\mathbb{R}} f \, d \ell \end{equation} |
| (2) | \begin{equation} \int_{\mathbb{R}} \left( f \circ T \right) d \ell = \frac{1}{|\text{det} T|} \int_{\mathbb{R}} f \, d \ell \end{equation} |