Linear Change of Variables for Lebesgue Measure on ℝⁿ

In the setting of Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation, assume the axiom of dependent choice. Let $n\in\mathbb{N}$, let $A\in\mathcal{M}_{n}(\mathbb{R})$ be invertible, let $b\in\mathbb{R}^{n}$, and let $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the function

$$T(x)=Ax+b .$$

(i) Images of Borel sets. For every $B\in\mathcal{B}(\mathbb{R}^{n})$, $T[B]\in\mathcal{B}(\mathbb{R}^{n})$ and

$$\lambda_{n}(T[B])=|\det A|\,\lambda_{n}(B).$$

(ii) The image measure. $T$ is $\mathcal{B}(\mathbb{R}^{n})/\mathcal{B}(\mathbb{R}^{n})$-measurable, and its image measure is

$$(\lambda_{n}\circ T^{-1})(B)=|\det A|^{-1}\lambda_{n}(B)\qquad(B\in\mathcal{B}(\mathbb{R}^{n})).$$

(iii) Nonnegative functions. For every $f\in\mathcal{M}^{+}(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$, $f\circ T\in\mathcal{M}^{+}(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$ and

$$\int f\circ T\,d\lambda_{n}=|\det A|^{-1}\int f\,d\lambda_{n}.$$

(iv) Integrable functions. Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be measurable. Then $f\circ T$ is measurable, $f\circ T\in\mathcal{L}^{1}(\lambda_{n})$ if and only if $f\in\mathcal{L}^{1}(\lambda_{n})$, and in that case

$$\int f\circ T\,d\lambda_{n}=|\det A|^{-1}\int f\,d\lambda_{n}.$$