ThmDex – An index of mathematical definitions, results, and conjectures.
Formulation F11545 on R4915:
F11545
Formulation 0
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}