Proof of Affine Change of Variables for Lebesgue Measure on the Real Line

Routine facts about real numbers are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, Toolkit for CA1c), and so is the agreement of the earlier and current versions of the Borel $\sigma$-algebra, measurable functions, measures, the integral of nonnegative functions and integrable functions (The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (ii), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (iv), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (iii), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (v), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (vi)). Throughout, write $c=|a|$, a positive real number since $a\neq0$. The plan is to show that the image of $\lambda$ under $T$, scaled by $c$, gives every half-open interval its length, so that it is $\lambda$ itself; the formulas for functions then follow from those for sets.

The image measure. The affine map $T$ is continuous, and continuous maps are Borel measurable (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv)), so $T$ is $\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R})$-measurable. Since $\lambda$ is a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (iv); this clause, like the uniqueness clause used in Preimages of Borel sets, rests on dependent choice), the image measure $\lambda\circ T^{-1}$, which assigns to a Borel set $B$ the value $\lambda(T^{-1}(B))$, is again a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (i)).

Scaling the image measure. Let $\nu$ be the measure with density the constant function $c$ with respect to $\lambda\circ T^{-1}$ (a nonnegative constant lies in $\mathcal{M}^{+}$ by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iii) and Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii)); it is a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (v)). To compute its values, write the integral over $B$ as the integral of $c\,\mathbf{1}_{B}$ (Integrals over Measurable Subsets (i)), pull out the constant (Basic Properties of the Integral of Nonnegative Measurable Maps (iv)) and integrate the indicator (Basic Properties of the Integral of Nonnegative Measurable Maps (ii)): for every $B\in\mathcal{B}(\mathbb{R})$,

$$\nu(B)=\int_{B}c\,d(\lambda\circ T^{-1})=c\int_{\mathbb{R}}\mathbf{1}_{B}\,d(\lambda\circ T^{-1})=c\,\lambda\bigl(T^{-1}(B)\bigr).$$

Preimages of Borel sets. We compare $\nu$ with $\lambda$ on half-open intervals. Let $u<v$. If $a>0$, then $u<ax+b\le v$ holds exactly when $(u-b)/a<x\le(v-b)/a$, so $T^{-1}((u,v])=((u-b)/a,(v-b)/a]$; if $a<0$, dividing by $a$ reverses the inequalities and $T^{-1}((u,v])=[(v-b)/a,(u-b)/a)$. In either case the preimage is an interval of length $(v-u)/c$, so $\lambda(T^{-1}((u,v]))=(v-u)/c$ (Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i)), and the formula of Scaling the image measure gives

$$\nu\bigl((u,v]\bigr)=c\cdot\frac{v-u}{c}=v-u .$$

Thus $\nu$ is a measure on the Borel sets that gives every half-open interval its length, and by the uniqueness of Lebesgue measure (Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (vi)) $\nu=\lambda$. In other words $c\,\lambda(T^{-1}(B))=\lambda(B)$ for every $B\in\mathcal{B}(\mathbb{R})$, and dividing by $c$ gives $\lambda(T^{-1}(B))=|a|^{-1}\lambda(B)$. Together with the measurability of $T$ shown in The image measure, this proves (i).

Nonnegative functions. Let $f\in\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$. By the change-of-variables formula for image measures (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (ii)), $f\circ T\in\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$ and

$$\int_{\mathbb{R}}f\circ T\,d\lambda=\int_{\mathbb{R}}f\,d(\lambda\circ T^{-1}).$$

To evaluate the right-hand side we recognise $\lambda\circ T^{-1}$ as a measure with density with respect to $\lambda$. By Preimages of Borel sets, $\lambda\circ T^{-1}$ assigns the value $c^{-1}\lambda(B)$ to each Borel set $B$; the computation of Scaling the image measure, carried out with $\lambda$ in place of $\lambda\circ T^{-1}$ and the constant $c^{-1}$ in place of $c$, shows that the measure with constant density $c^{-1}$ with respect to $\lambda$ (again a nonnegative constant in $\mathcal{M}^{+}$, by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iii) and Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii)) assigns the same value. So $\lambda\circ T^{-1}$ is that measure, and integrating against a density (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (vi)) and then pulling out the constant (Basic Properties of the Integral of Nonnegative Measurable Maps (iv)) give

$$\int_{\mathbb{R}}f\,d(\lambda\circ T^{-1})=\int_{\mathbb{R}}c^{-1}f\,d\lambda=c^{-1}\int_{\mathbb{R}}f\,d\lambda .$$

Combining the two displays proves (ii).

Integrable functions. Let $f:\mathbb{R}\to\mathbb{R}$ be measurable. Then $f\circ T$ is measurable as the composite of the measurable maps $T$ and $f$ (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (ii)). Taking positive and negative parts and absolute values commutes with composition: pointwise, $(f\circ T)^{\pm}=f^{\pm}\circ T$ and $|f\circ T|=|f|\circ T$. Applying Nonnegative functions to $|f|$ gives

$$\int_{\mathbb{R}}|f\circ T|\,d\lambda=\int_{\mathbb{R}}|f|\circ T\,d\lambda=c^{-1}\int_{\mathbb{R}}|f|\,d\lambda,$$

so one side is finite exactly when the other is, and the integrability criterion (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (iv)) shows that $f\circ T\in\mathcal{L}^{1}(\lambda)$ if and only if $f\in\mathcal{L}^{1}(\lambda)$. In that case the integral of an integrable function is the difference of the integrals of its positive and negative parts (Integrable Functions and Their Lebesgue Integral (iii)), and Nonnegative functions applied to $f^{+}$ and to $f^{-}$ gives

$$\int_{\mathbb{R}}f\circ T\,d\lambda=\int_{\mathbb{R}}f^{+}\circ T\,d\lambda-\int_{\mathbb{R}}f^{-}\circ T\,d\lambda=c^{-1}\Bigl(\int_{\mathbb{R}}f^{+}\,d\lambda-\int_{\mathbb{R}}f^{-}\,d\lambda\Bigr)=c^{-1}\int_{\mathbb{R}}f\,d\lambda,$$

which completes the proof of (iii).