Proof of Lebesgue Measure on ℝⁿ: σ-Finiteness, Bounded Sets, Open Sets and Coordinate Hyperplanes

Open boxes lie in $\mathcal{B}(\mathbb{R}^{n})$ (Existence and Uniqueness of Lebesgue Measure on ℝⁿ (i), Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii)), and the cubes $Q_{k}$, $k\in\mathbb{N}$, cover $\mathbb{R}^{n}$ (Existence and Uniqueness of Lebesgue Measure on ℝⁿ (ii)). For $a,b\in\mathbb{R}^{n}$ with $a_{i}<b_{i}$ for every $i\in[n]$,

$$\lambda_{n}\bigl(R(a,b)\bigr)=\prod_{i=1}^{n}(b_{i}-a_{i})$$

(Lebesgue Measure on $\mathbb{R}^n$ (i), Existence and Uniqueness of Lebesgue Measure on ℝⁿ (iii)); in particular $\lambda_{n}(Q_{k})=\prod_{i=1}^{n}2k$ is a real number, so $\lambda_{n}(Q_{k})<\infty$.

$\sigma$-finiteness. $(Q_{k})_{k\in\mathbb{N}}$ is a sequence in $\mathcal{B}(\mathbb{R}^{n})$ with union $\mathbb{R}^{n}$ and $\lambda_{n}(Q_{k})<\infty$ for every $k$, so $\lambda_{n}$ is $\sigma$-finite (Measure, Measure Space, and Probability Measure (iv)). This proves (i).

Bounded sets. Let $B\in\mathcal{B}(\mathbb{R}^{n})$ be bounded, with $d_{E}(x,y)\le M$ for all $x,y\in B$ (Bounded Subset of a Metric Space (i)). If $B=\emptyset$, then $\lambda_{n}(B)=0$. Otherwise fix $x^{0}\in B$ and choose $k\in\mathbb{N}$ with $M+\lVert x^{0}\rVert<k$ (The Integers in the Real Numbers: Greatest Elements, the Integer Part, and Archimedean Consequences (ii)). For $x\in B$ and $i\in[n]$,

$$|x_{i}|\le\lVert x\rVert\le\lVert x-x^{0}\rVert+\lVert x^{0}\rVert=d_{E}(x,x^{0})+\lVert x^{0}\rVert\le M+\lVert x^{0}\rVert<k$$

(Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iv), Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (vi), Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (ii)). So $B\subseteq Q_{k}$, and $\lambda_{n}(B)\le\lambda_{n}(Q_{k})<\infty$ (Basic Properties of a Measure (iii)). This proves (ii).

Open sets. Let $U\subseteq\mathbb{R}^{n}$ be open and nonempty. Then $U\in\mathcal{B}(\mathbb{R}^{n})$ (Borel Sigma-Algebra of a Metric Space (i), Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii)). Fix $x\in U$. By The Borel Sets of ℝⁿ and of the Real Matrices Are Generated by the Coordinates (i), $U$ is the union of a sequence of rational open boxes contained in $U$, so $x\in W\subseteq U$ for one of them, $W=R(a,b)$ with $a,b$ having rational, hence real, components. As $a_{i}<x_{i}<b_{i}$, we have $b_{i}-a_{i}>0$ for every $i\in[n]$, so the product $\prod_{i=1}^{n}(b_{i}-a_{i})$ is nonnegative and nonzero (Products over Finite Sets: a Single Factor, Vanishing in a Field, and Signs and Monotonicity in an Ordered Ring (iii), Products over Finite Sets: a Single Factor, Vanishing in a Field, and Signs and Monotonicity in an Ordered Ring (ii)). Hence

$$\lambda_{n}(U)\ge\lambda_{n}(W)=\prod_{i=1}^{n}(b_{i}-a_{i})>0$$

(Basic Properties of a Measure (iii)). This proves (iii).

Coordinate hyperplanes. The set $H$ is the preimage of $\{c\}=[c,c]\in\mathcal{B}(\mathbb{R})$ under the $i$-th coordinate function, so $H\in\mathcal{B}(\mathbb{R}^{n})$ (Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (ii), The Borel Sets of ℝⁿ and of the Real Matrices Are Generated by the Coordinates (iii), Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii)), and so is each $H\cap Q_{k}$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (ix)). Fix $k\in\mathbb{N}$ and put $C=\prod_{l=1}^{n}2k$, a positive real number (Products over Finite Sets: a Single Factor, Vanishing in a Field, and Signs and Monotonicity in an Ordered Ring (iii), Products over Finite Sets: a Single Factor, Vanishing in a Field, and Signs and Monotonicity in an Ordered Ring (ii)). For $j\in\mathbb{N}$ let $a,b\in\mathbb{R}^{n}$ have components $a_{l}=-k$, $b_{l}=k$ for $l\neq i$, and $a_{i}=c-k/j$, $b_{i}=c+k/j$. Every $x\in H\cap Q_{k}$ has $x_{i}=c$ and $-k<x_{l}<k$ for all $l$, so $H\cap Q_{k}\subseteq R(a,b)$. Moreover $b_{l}-a_{l}=2k\,w_{l}$, where $w_{l}=1$ for $l\neq i$ and $w_{i}=1/j$, so

$$\lambda_{n}(H\cap Q_{k})\le\lambda_{n}\bigl(R(a,b)\bigr)=\prod_{l=1}^{n}2k\,w_{l}=\Bigl(\prod_{l=1}^{n}2k\Bigr)\Bigl(\prod_{l=1}^{n}w_{l}\Bigr)=\frac{C}{j}$$

(Basic Properties of a Measure (iii), Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (v); $\prod_{l=1}^{n}w_{l}=w_{i}=1/j$ by Products over Finite Sets: a Single Factor, Vanishing in a Field, and Signs and Monotonicity in an Ordered Ring (i), all other factors being $1$). In particular $u=\lambda_{n}(H\cap Q_{k})\le C/j$ is a real number. If $u$ were positive, there would be $j\in\mathbb{N}$ with $1/j<u/C$ (The Integers in the Real Numbers: Greatest Elements, the Integer Part, and Archimedean Consequences (iii)), contradicting $u\le C/j$. So $\lambda_{n}(H\cap Q_{k})=0$ for every $k$, and since $H=\bigcup_{k\in\mathbb{N}}(H\cap Q_{k})$,

$$\lambda_{n}(H)=0$$

(Null Sets and Properties Holding Almost Everywhere (iv)). This proves (iv).