Proof of Existence and Uniqueness of Lebesgue Measure on ℝⁿ

Steps 1 to 4 are proved for an arbitrary dimension $n\in\mathbb{N}$, and are used in Step 6 in every dimension.

Step 1: open boxes are Borel. Let $a,b\in\mathbb{R}^{n}$. For $i\in[n]$, the set $\{x\in\mathbb{R}^{n}:a_{i}<x_{i}<b_{i}\}$ is the preimage of the open interval $(a_{i},b_{i})\in\mathcal{B}(\mathbb{R})$ under the $i$-th coordinate function, so it lies 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)). The open box $R(a,b)$ is the intersection of these $n$ sets, a finite intersection of Borel sets, so $R(a,b)\in\mathcal{B}(\mathbb{R}^{n})$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (ix), by induction on the number of sets). Hence $\mathcal{R}_{n}\subseteq\mathcal{B}(\mathbb{R}^{n})$.

Step 2: a $\pi$-system. Let $a,b,a',b'\in\mathbb{R}^{n}$. By Open Boxes: Emptiness, Intersections and Preimages under Concatenation (ii), the points $c,d\in\mathbb{R}^{n}$ with components $c_{i}=\max\{a_{i},a'_{i}\}$ and $d_{i}=\min\{b_{i},b'_{i}\}$ satisfy

$$R(a,b)\cap R(a',b')=R(c,d)\in\mathcal{R}_{n},$$

and $\mathcal{R}_{n}$ is a $\pi$-system on $\mathbb{R}^{n}$.

Step 3: the boxes generate $\mathcal{B}(\mathbb{R}^{n})$. By Step 1, $\sigma(\mathcal{R}_{n})\subseteq\mathcal{B}(\mathbb{R}^{n})$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv)). Conversely, let $U\subseteq\mathbb{R}^{n}$ be open. By The Borel Sets of ℝⁿ and of the Real Matrices Are Generated by the Coordinates (i), $U=\bigcup_{k\in\mathbb{N}}W_{k}$, where each $W_{k}$ is a rational open box $R(a,b)$, $a,b\in\mathbb{Q}^{n}$; reading the components of $a$ and $b$ as real numbers (The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (v)), $W_{k}$ is an open box in the sense of Open Boxes in ℝⁿ (i), so $W_{k}\in\mathcal{R}_{n}$ and $U\in\sigma(\mathcal{R}_{n})$. Thus every open subset of $\mathbb{R}^{n}$ lies in $\sigma(\mathcal{R}_{n})$, and $\mathcal{B}(\mathbb{R}^{n})\subseteq\sigma(\mathcal{R}_{n})$ (Borel Sigma-Algebra of a Metric Space (i), Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv)).

Step 4: the cubes cover $\mathbb{R}^{n}$. Let $x\in\mathbb{R}^{n}$. There is $k\in\mathbb{N}$ with $\lVert x\rVert<k$ (The Integers in the Real Numbers: Greatest Elements, the Integer Part, and Archimedean Consequences (ii), with $\varepsilon=1$), and $|x_{i}|\le\lVert x\rVert$ for every $i\in[n]$ (Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iv)). So $-k<x_{i}<k$ for every $i\in[n]$, that is, $x\in Q_{k}$. Steps 2 and 3 prove (i), and Step 4 proves (ii).

Step 5: boxes determine measures. Let $\mu$ and $\nu$ be as in (iv). The cubes $Q_{k}$ are open boxes, so $(Q_{k})_{k\in\mathbb{N}}$ is a sequence in $\mathcal{R}_{n}$ with union $\mathbb{R}^{n}$ (Step 4) and $\mu(Q_{k})<\infty$ for every $k$. Since $\mathcal{R}_{n}$ is a $\pi$-system generating $\mathcal{B}(\mathbb{R}^{n})$ on which $\mu$ and $\nu$ agree, $\mu=\nu$ by Uniqueness of Measures Agreeing on a Generating π-System (ii). This proves (iv).

Step 6: existence. For $p\in\mathbb{N}$ let $\varphi(p)$ be the statement: there is a measure $\kappa$ on $(\mathbb{R}^{p},\mathcal{B}(\mathbb{R}^{p}))$ with $\kappa(R(a,b))=\prod_{i=1}^{p}(b_{i}-a_{i})$ for all $a,b\in\mathbb{R}^{p}$ with $a_{i}<b_{i}$ for every $i\in[p]$. We prove $\varphi(p)$ for every $p\in\mathbb{N}$ by induction (Induction for Predicative Formulas, and the Order of ℕ below 1 and below n+1 (i); $\varphi$ is predicative by Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices (iv)). Throughout, products of nonnegative real numbers in $[0,\infty]$ are their real products (Addition and Multiplication on [0,∞] (ii)).

A measure $\kappa$ as in $\varphi(p)$ is $\sigma$-finite: the cubes $Q^{(p)}_{k}=R(-k\mathbf{u},k\mathbf{u})$ of $\mathbb{R}^{p}$, $k\in\mathbb{N}$, with $\mathbf{u}=(1,\dots,1)\in\mathbb{R}^{p}$, lie in $\mathcal{B}(\mathbb{R}^{p})$ (Step 1), cover $\mathbb{R}^{p}$ (Step 4), and $\kappa(Q^{(p)}_{k})=\prod_{i=1}^{p}2k$ is a real number.

Dimension one. Let $\tau:\mathbb{R}\to\mathbb{R}^{1}$ be the function with $\tau(t)_{1}=t$, which is $\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R}^{1})$-measurable by Borel Sets of ℝ^{m+n} and the Product σ-Algebra of ℝ^m and ℝ^n (iii). Lebesgue measure $\lambda$ is a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (iv)), so its image measure $\kappa_{1}=\lambda\circ\tau^{-1}$ is a measure on $(\mathbb{R}^{1},\mathcal{B}(\mathbb{R}^{1}))$ (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (i)). Let $a,b\in\mathbb{R}^{1}$ with $a_{1}<b_{1}$. Since $\tau(t)_{1}=t$, we have $\tau^{-1}(R(a,b))=(a_{1},b_{1})$, and

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

(Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i), Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (i)). So $\varphi(1)$ holds.

From $p$ to $p+1$. Assume $\varphi(p)$, let $\kappa_{p}$ be a measure as there, and let $\kappa_{1}$ be the measure just constructed; both are $\sigma$-finite. Their product measure $\kappa_{p}\otimes\kappa_{1}$ is a measure on the product $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{p})\otimes\mathcal{B}(\mathbb{R}^{1})$ with

$$(\kappa_{p}\otimes\kappa_{1})(A\times B)=\kappa_{p}(A)\,\kappa_{1}(B)\qquad(A\in\mathcal{B}(\mathbb{R}^{p}),\ B\in\mathcal{B}(\mathbb{R}^{1}))$$

(Existence and Uniqueness of the Product Measure (i)). The concatenation function $\iota:\mathbb{R}^{p}\times\mathbb{R}^{1}\to\mathbb{R}^{p+1}$ of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (ii) is $\mathcal{B}(\mathbb{R}^{p})\otimes\mathcal{B}(\mathbb{R}^{1})/\mathcal{B}(\mathbb{R}^{p+1})$-measurable (Borel Sets of ℝ^{m+n} and the Product σ-Algebra of ℝ^m and ℝ^n (i)), so $\kappa_{p+1}=(\kappa_{p}\otimes\kappa_{1})\circ\iota^{-1}$ is a measure on $(\mathbb{R}^{p+1},\mathcal{B}(\mathbb{R}^{p+1}))$ (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (i)).

Let $a,b\in\mathbb{R}^{p+1}$ with $a_{i}<b_{i}$ for every $i\in[p+1]$, and let $(a',a'')$ and $(b',b'')$ be the pairs that $\iota$ maps to $a$ and $b$ (Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (iii)): $a'_{i}=a_{i}$ and $b'_{i}=b_{i}$ for $i\in[p]$, and $a''_{1}=a_{p+1}$, $b''_{1}=b_{p+1}$. By Open Boxes: Emptiness, Intersections and Preimages under Concatenation (iii),

$$\iota^{-1}\bigl(R(a,b)\bigr)=R(a',b')\times R(a'',b''),$$

a product of Borel sets by Step 1. Here $\kappa_{p}(R(a',b'))=\prod_{i=1}^{p}(b_{i}-a_{i})$ by $\varphi(p)$, and $\kappa_{1}(R(a'',b''))=b''_{1}-a''_{1}=b_{p+1}-a_{p+1}$ by the dimension-one computation above. Therefore

$$\kappa_{p+1}\bigl(R(a,b)\bigr)=\kappa_{p}\bigl(R(a',b')\bigr)\,\kappa_{1}\bigl(R(a'',b'')\bigr)=\Bigl(\prod_{i=1}^{p}(b_{i}-a_{i})\Bigr)(b_{p+1}-a_{p+1})=\prod_{i=1}^{p+1}(b_{i}-a_{i}),$$

the last equality by Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (i). So $\varphi(p+1)$ holds, and by induction $\varphi(n)$ holds: a measure as in (iii) exists.

Step 7: uniqueness. Let $\kappa$ and $\kappa'$ be measures as in (iii), and let $a,b\in\mathbb{R}^{n}$. If $a_{i}<b_{i}$ for every $i\in[n]$, then $\kappa(R(a,b))=\prod_{i=1}^{n}(b_{i}-a_{i})=\kappa'(R(a,b))$, a real number. Otherwise $b_{i}\le a_{i}$ for some $i$, so $R(a,b)=\emptyset$ (Open Boxes: Emptiness, Intersections and Preimages under Concatenation (i)) and $\kappa(R(a,b))=0=\kappa'(R(a,b))$. Thus $\kappa$ and $\kappa'$ agree and are finite on every open box, and $\kappa=\kappa'$ by Step 5.