Proof of The Lebesgue Integral over ℝⁿ of a Product of Functions of One Coordinate Each

Throughout, composites and products of measurable functions into $[0,\infty]$ are measurable, and the product of $[0,\infty]$ is associative and commutative (all background of the setting). For $p\in\mathbb{N}$ and $h_{1},\dots,h_{p}$ write $F_{p}(x)=\prod_{j=1}^{p}h_{j}(x_{j})$ for $x\in\mathbb{R}^{p}$.

Step 1 (dimension one). Let $g\in\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$. The function $\tau:\mathbb{R}\to\mathbb{R}^{1}$ of Borel Sets of ℝ^{m+n} and the Product σ-Algebra of ℝ^m and ℝ^n (iii) is a bijection whose inverse $\tau^{-1}:\eta\mapsto\eta_{1}$ is $\mathcal{B}(\mathbb{R}^{1})/\mathcal{B}(\mathbb{R})$-measurable. For $B\in\mathcal{B}(\mathbb{R})$ we have $(\tau^{-1})^{-1}[B]=\tau[B]$, so $\lambda_{1}((\tau^{-1})^{-1}[B])=\lambda(B)$ by Lebesgue Measure on ℝ^{m+n} as the Product of Lebesgue Measures on ℝ^m and ℝ^n (i); that is, the image measure of $\lambda_{1}$ under $\tau^{-1}$ is

$$\lambda_{1}\circ(\tau^{-1})^{-1}=\lambda .$$

The change of variables Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (ii) along $\tau^{-1}$, applied to $g$, therefore gives

$$\int_{\mathbb{R}^{1}}g(\eta_{1})\,d\lambda_{1}(\eta)=\int_{\mathbb{R}}g\,d(\lambda_{1}\circ(\tau^{-1})^{-1})=\int_{\mathbb{R}}g\,d\lambda .$$

Step 2 (proof of (i)). We prove the clause, measurability of $F_{p}$ included, for all $h_{1},\dots,h_{p}$ by induction on $p$ (Induction for Predicative Formulas, and the Order of ℕ below 1 and below n+1 (i); the condition on $p$ is predicative by The Borel Sets of ℝⁿ, Lebesgue Measure on ℝⁿ and Integrals against It Are Predicative in the Dimension (iii), The Simple, Nonnegative, Integrable and Signed Integrals Are Predicative in the Measure and the Function, at Any Variable Positions (ii), and the formula trees of Formula Trees with Values: Syntax, Two Readings, and Definedness (i) with The Graph Reading of a Formula Tree Is Predicative and Agrees with Its Reading, So the Reading Is Predicative (i), The Graph Reading of a Formula Tree Is Predicative and Agrees with Its Reading, So the Reading Is Predicative (ii) and Predicative Formulas for the Vocabulary of Finite Combinatorics, at Any Variable Positions (ii), Predicative Formulas for the Vocabulary of Finite Combinatorics, at Any Variable Positions (iv), Predicative Formulas for the Vocabulary of Finite Combinatorics, at Any Variable Positions (vi)). For $p=1$, $F_{1}(x)=h_{1}(x_{1})$ by Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (i), that is, $F_{1}=h_{1}\circ\tau^{-1}$, which is measurable because $\tau^{-1}$ is (Step 1); by the same clause $\prod_{j=1}^{1}\int_{\mathbb{R}}h_{j}\,d\lambda=\int_{\mathbb{R}}h_{1}\,d\lambda$, so Step 1 with $g=h_{1}$ gives the claim. For the step from $p$ to $p+1$, let $h_{1},\dots,h_{p+1}\in\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$; the induction hypothesis is applied to $h_{1},\dots,h_{p}$, the restriction of $j\mapsto h_{j}$ to $[p]$, and gives that $F_{p}$ is measurable and satisfies the clause. Let $\iota:\mathbb{R}^{p}\times\mathbb{R}^{1}\to\mathbb{R}^{p+1}$ be the concatenation function; $\iota(\xi,\eta)_{j}=\xi_{j}$ for $j\in[p]$ and $\iota(\xi,\eta)_{p+1}=\eta_{1}$ (Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (ii)), so by Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (i)

$$F_{p+1}(\iota(\xi,\eta))=F_{p}(\xi)\cdot h_{p+1}(\eta_{1}),$$

so that $F_{p+1}(x)=F_{p}(\xi)\cdot h_{p+1}(x_{p+1})$ for $x\in\mathbb{R}^{p+1}$ with $\iota^{-1}(x)=(\xi,\eta)$. With $\pi_{1}$ and $\pi_{2}$ the projections of $\mathbb{R}^{p}\times\mathbb{R}^{1}$ onto its factors, this says

$$F_{p+1}=(F_{p}\circ\pi_{1}\circ\iota^{-1})\cdot(h_{p+1}\circ\tau^{-1}\circ\pi_{2}\circ\iota^{-1}),$$

a product of composites of measurable functions: $\iota^{-1}$ is measurable by Borel Sets of ℝ^{m+n} and the Product σ-Algebra of ℝ^m and ℝ^n (ii), $\pi_{1}$ and $\pi_{2}$ by The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (vi), $\tau^{-1}$ by Step 1, and $F_{p}$ by the induction hypothesis. So $F_{p+1}$ is measurable by the opening paragraph, which is the measurability part of the claim for $p+1$. By Tonelli's formula Lebesgue Measure on ℝ^{m+n} as the Product of Lebesgue Measures on ℝ^m and ℝ^n (iv), with dimensions $p$ and $1$, then The Integral of a Multiple by a Constant in [0,∞], the Constant ∞ Included (i) with $c=F_{p}(\xi)$ and Step 1 for the inner integral, The Integral of a Multiple by a Constant in [0,∞], the Constant ∞ Included (i) with $c=\int_{\mathbb{R}}h_{p+1}\,d\lambda$ for the outer one, and the induction hypothesis,

$$\int_{\mathbb{R}^{p+1}}F_{p+1}\,d\lambda_{p+1}=\int_{\mathbb{R}^{p}}F_{p}(\xi)\cdot\Bigl(\int_{\mathbb{R}}h_{p+1}\,d\lambda\Bigr)d\lambda_{p}(\xi)=\Bigl(\prod_{j=1}^{p}\int_{\mathbb{R}}h_{j}\,d\lambda\Bigr)\cdot\int_{\mathbb{R}}h_{p+1}\,d\lambda ,$$

which is $\prod_{j=1}^{p+1}\int_{\mathbb{R}}h_{j}\,d\lambda$ by Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (i).

Step 3 (proof of (ii)). By Integrals over Measurable Subsets (i), $h_{j}\mathbf{1}_{B_{j}}\in\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$ and $\int_{\mathbb{R}}h_{j}\mathbf{1}_{B_{j}}\,d\lambda=\int_{B_{j}}h_{j}\,d\lambda$ for every $j\in[n]$. Let $G(x)=\prod_{j=1}^{n}(h_{j}\mathbf{1}_{B_{j}})(x_{j})$ for $x\in\mathbb{R}^{n}$ be the integrand of (ii). Then $G$ equals the function that (i) gives for the family $j\mapsto h_{j}\mathbf{1}_{B_{j}}$, as the two agree at every $x$, and (i) for this family gives the claim.