Affine Change of Variables for Lebesgue Measure on the Real Line

In the setting of Measurable Functions and the Lebesgue Integral: Notation, let $\lambda$ be Lebesgue measure on $\mathbb{R}$, let $a,b\in\mathbb{R}$ with $a\neq0$, and let $T:\mathbb{R}\to\mathbb{R}$ be the function

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

(i) Preimages of Borel sets. $T$ is $\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R})$-measurable, and

$$\lambda\bigl(T^{-1}(B)\bigr)=|a|^{-1}\,\lambda(B)\qquad(B\in\mathcal{B}(\mathbb{R})).$$

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

$$\int_{\mathbb{R}}f(ax+b)\,d\lambda(x)=|a|^{-1}\int_{\mathbb{R}}f\,d\lambda .$$

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

$$\int_{\mathbb{R}}f(ax+b)\,d\lambda(x)=|a|^{-1}\int_{\mathbb{R}}f\,d\lambda .$$