Proof of Lebesgue Measure on ℝ^{m+n} as the Product of Lebesgue Measures on ℝ^m and ℝ^n

Throughout, products of nonnegative real numbers in $[0,\infty]$ are their real products (Addition and Multiplication on [0,∞] (ii)). A measure on $(\mathbb{R}^{p},\mathcal{B}(\mathbb{R}^{p}))$, $p\in\mathbb{N}$, that gives every open box $R(a,b)$ with $a_{i}<b_{i}$ for all $i\in[p]$ the value $\prod_{i=1}^{p}(b_{i}-a_{i})$ is $\lambda_{p}$, by the uniqueness in Existence and Uniqueness of Lebesgue Measure on ℝⁿ (iii) and Lebesgue Measure on $\mathbb{R}^n$ (i).

Dimension one. By Borel Sets of ℝ^{m+n} and the Product σ-Algebra of ℝ^m and ℝ^n (iii), $\tau$ is a $\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R}^{1})$-measurable bijection, and $\tau[B]\in\mathcal{B}(\mathbb{R}^{1})$ for every $B\in\mathcal{B}(\mathbb{R})$. The image measure $\lambda\circ\tau^{-1}$ 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), Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (iv)). For $a,b\in\mathbb{R}^{1}$ with $a_{1}<b_{1}$ we have $\tau^{-1}(R(a,b))=(a_{1},b_{1})$, since $\tau(t)_{1}=t$, so

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

(Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i), Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (i)). Hence $\lambda\circ\tau^{-1}=\lambda_{1}$, and for every $B\in\mathcal{B}(\mathbb{R})$, since $\tau^{-1}(\tau[B])=B$ for the bijection $\tau$ (Images and Preimages under a Bijection and Its Inverse (ii)),

$$\lambda_{1}(\tau[B])=\lambda\bigl(\tau^{-1}(\tau[B])\bigr)=\lambda(B).$$

This proves (i).

Concatenation. The measures $\lambda_{m}$ and $\lambda_{n}$ are $\sigma$-finite (Lebesgue Measure on ℝⁿ: σ-Finiteness, Bounded Sets, Open Sets and Coordinate Hyperplanes (i)), so $\lambda_{m}\otimes\lambda_{n}$ is a measure on $\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n})$ with

$$(\lambda_{m}\otimes\lambda_{n})(A\times B)=\lambda_{m}(A)\,\lambda_{n}(B)\qquad(A\in\mathcal{B}(\mathbb{R}^{m}),\ B\in\mathcal{B}(\mathbb{R}^{n}))$$

(Existence and Uniqueness of the Product Measure (i)). Since $\iota$ is $\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n})/\mathcal{B}(\mathbb{R}^{m+n})$-measurable (Borel Sets of ℝ^{m+n} and the Product σ-Algebra of ℝ^m and ℝ^n (i)), $\rho=(\lambda_{m}\otimes\lambda_{n})\circ\iota^{-1}$ is a measure on $(\mathbb{R}^{m+n},\mathcal{B}(\mathbb{R}^{m+n}))$ (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (i)).

Let $a,b\in\mathbb{R}^{m+n}$ with $a_{k}<b_{k}$ for every $k\in[m+n]$, 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)): $a'_{k}=a_{k}$ and $b'_{k}=b_{k}$ for $k\in[m]$, and $a''_{j}=a_{m+j}$ and $b''_{j}=b_{m+j}$ for $j\in[n]$. By Open Boxes: Emptiness, Intersections and Preimages under Concatenation (iii),

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

where $R(a',b')\in\mathcal{B}(\mathbb{R}^{m})$ and $R(a'',b'')\in\mathcal{B}(\mathbb{R}^{n})$, open boxes being Borel (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)). The product formula therefore applies, and

$$\rho\bigl(R(a,b)\bigr)=\lambda_{m}\bigl(R(a',b')\bigr)\,\lambda_{n}\bigl(R(a'',b'')\bigr)=\Bigl(\prod_{k=1}^{m}(b_{k}-a_{k})\Bigr)\Bigl(\prod_{j=1}^{n}(b_{m+j}-a_{m+j})\Bigr)=\prod_{k=1}^{m+n}(b_{k}-a_{k}),$$

the last equality by Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (iii). Therefore $\rho=\lambda_{m+n}$. This proves (ii).

Images of product-measurable sets. For $E\in\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n})$, $\iota[E]\in\mathcal{B}(\mathbb{R}^{m+n})$ by Borel Sets of ℝ^{m+n} and the Product σ-Algebra of ℝ^m and ℝ^n (ii), and $\iota^{-1}(\iota[E])=E$ because $\iota$ is a bijection (Images and Preimages under a Bijection and Its Inverse (ii)); so, by (ii),

$$\lambda_{m+n}(\iota[E])=\rho(\iota[E])=(\lambda_{m}\otimes\lambda_{n})(E).$$

This proves (iii).

Nonnegative functions. Let $f\in\mathcal{M}^{+}(\mathbb{R}^{m+n},\mathcal{B}(\mathbb{R}^{m+n}))$ and $F=f\circ\iota$, so $F(\xi,\eta)=f(\iota(\xi,\eta))$. By Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (ii) and (ii), $F\in\mathcal{M}^{+}(\mathbb{R}^{m}\times\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n}))$ and

$$\int_{\mathbb{R}^{m+n}}f\,d\lambda_{m+n}=\int_{\mathbb{R}^{m+n}}f\,d\bigl((\lambda_{m}\otimes\lambda_{n})\circ\iota^{-1}\bigr)=\int_{\mathbb{R}^{m}\times\mathbb{R}^{n}}F\,d(\lambda_{m}\otimes\lambda_{n}).$$

Tonelli's theorem Tonelli's and Fubini's Theorems (i), applied to $F$ and the $\sigma$-finite measures $\lambda_{m}$ and $\lambda_{n}$, says that each function $\eta\mapsto F(\xi,\eta)$ lies in $\mathcal{M}^{+}(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$, that $\xi\mapsto\int_{\mathbb{R}^{n}}F(\xi,\eta)\,d\lambda_{n}(\eta)$ lies in $\mathcal{M}^{+}(\mathbb{R}^{m},\mathcal{B}(\mathbb{R}^{m}))$, and that the last integral above equals $\int_{\mathbb{R}^{m}}\bigl(\int_{\mathbb{R}^{n}}F(\xi,\eta)\,d\lambda_{n}(\eta)\bigr)d\lambda_{m}(\xi)$. This is (iv).

Integrable functions. Let $f\in\mathcal{L}^{1}(\lambda_{m+n})$ and $F=f\circ\iota$. By Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (iii) and (ii), $F\in\mathcal{L}^{1}(\lambda_{m}\otimes\lambda_{n})$ and

$$\int_{\mathbb{R}^{m+n}}f\,d\lambda_{m+n}=\int_{\mathbb{R}^{m}\times\mathbb{R}^{n}}F\,d(\lambda_{m}\otimes\lambda_{n}).$$

The section of $F$ at $\xi\in\mathbb{R}^{m}$ is the function $\eta\mapsto f(\iota(\xi,\eta))$ (Precomposition Commutes with Pointwise Operations, and Sections of Functions Are Precompositions (iv)), so the set of $\xi$ at which this section does not lie in $\mathcal{L}^{1}(\lambda_{n})$ is the set $N$ of (v), and the function $I$ of Tonelli's and Fubini's Theorems (ii) for $F$ is the function $g$ of (v). Fubini's theorem Tonelli's and Fubini's Theorems (ii), applied to $F$ and the $\sigma$-finite measures $\lambda_{m}$ and $\lambda_{n}$, therefore gives $N\in\mathcal{B}(\mathbb{R}^{m})$, $\lambda_{m}(N)=0$ and $g\in\mathcal{L}^{1}(\lambda_{m})$, with

$$\int_{\mathbb{R}^{m}\times\mathbb{R}^{n}}F\,d(\lambda_{m}\otimes\lambda_{n})=\int_{\mathbb{R}^{m}}g\,d\lambda_{m}.$$

This proves (v).