Proof of Lebesgue Measure on a Concatenated Euclidean Space: Iterated Integration over the Two Factors

Each result cited is universally quantified over the data in its own statement. Throughout, $\lambda$ is Lebesgue measure on $\mathcal{B}(\mathbb{R})$ and, as in Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ and Lebesgue Measure on $\mathbb{R}^n$, points of $\mathbb{R}^{1}$ are identified with real numbers, so that $\mathcal{B}(\mathbb{R}^{1})=\mathcal{B}_{1}=\mathcal{B}(\mathbb{R})$ and $\lambda_{1}=\lambda$. For every $m\in\mathbb{N}$ the measure $\lambda_{m}$ is $\sigma$-finite by Lebesgue Measure on Euclidean Space is Sigma-Finite §sigma-finite; hence for $m,n\in\mathbb{N}$ the product measure $\lambda_{m}\otimes\lambda_{n}$ on $\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n})$ exists by Existence and Uniqueness of the Product Measure, and Tonelli and Fubini Theorems applies to the pair $\lambda_{m},\lambda_{n}$.

Step 1 (Composition with a concatenation). Let $m,n\in\mathbb{N}$ and let $F:\mathbb{R}^{m+n}\to[0,\infty]$ be Borel. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product the map $\iota^{m,n}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n})$ and $\mathcal{B}(\mathbb{R}^{m+n})$. For real $c$ the set $\{F\circ\iota^{m,n}>c\}$ is the preimage under $\iota^{m,n}$ of $\{F>c\}\in\mathcal{B}(\mathbb{R}^{m+n})$, so $F\circ\iota^{m,n}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n})$ in the sense of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable. By the Sections and Tonelli parts of Tonelli and Fubini Theorems, for every $u\in\mathbb{R}^{m}$ the map $v\mapsto F(\iota^{m,n}(u,v))$ is Borel on $\mathbb{R}^{n}$, the map $u\mapsto\int_{\mathbb{R}^{n}}F(\iota^{m,n}(u,v))\,\lambda_{n}(dv)$ is Borel on $\mathbb{R}^{m}$, and

It therefore suffices to show that for all $m,n\in\mathbb{N}$ and every Borel $F:\mathbb{R}^{m+n}\to[0,\infty]$

because (1) and (2) with $m=q$, $n=p$ and $F=f$ give all assertions of the lemma. Let $S$ be the set of $n\in\mathbb{N}$ such that (2) holds for every $m\in\mathbb{N}$ and every Borel $F:\mathbb{R}^{m+n}\to[0,\infty]$. We show $1\in S$ and that $n\in S$ implies $n+1\in S$; since $n+1$ is the successor of $n$ by claim 1 of Arithmetic of Addition on the Natural Numbers, Principle of Induction for the Natural Numbers then gives $S=\mathbb{N}$.

Step 2 ($1\in S$). Let $m\in\mathbb{N}$. Since $m+1$ is the successor of $m$ and $m+1\ne1$ (claims 1 and 7 of Arithmetic of Addition on the Natural Numbers), Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ with $l=m+1$ defines $\mathcal{B}_{m+1}=\mathcal{B}_{m}\otimes\mathcal{B}(\mathbb{R})$ and $\lambda_{m+1}=\lambda_{m}\otimes\lambda$, transported to $\mathbb{R}^{m+1}$ along the bijection $J:\mathbb{R}^{m}\times\mathbb{R}\to\mathbb{R}^{m+1}$, $J((\theta_{1},\dots,\theta_{m}),t)=(\theta_{1},\dots,\theta_{m},t)$; thus $E\in\mathcal{B}_{m+1}$ exactly when $J^{-1}(E)\in\mathcal{B}_{m}\otimes\mathcal{B}(\mathbb{R})$, and then $\lambda_{m+1}(E)=(\lambda_{m}\otimes\lambda)(J^{-1}(E))$. By Lebesgue Measure on $\mathbb{R}^n$, $\mathcal{B}(\mathbb{R}^{k})=\mathcal{B}_{k}$ and Lebesgue measure on it is the measure $\lambda_{k}$ of that lemma, for $k=m$ and $k=m+1$. By the definition of the concatenation in Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space, $\iota^{m,1}(u,t)$ has $k$th coordinate $u_{k}$ for $k\in[m]$ and $(m+1)$th coordinate $t$, so $\iota^{m,1}=J$. Hence $\lambda_{m+1}(E)=(\lambda_{m}\otimes\lambda_{1})\bigl((\iota^{m,1})^{-1}(E)\bigr)$ for every $E\in\mathcal{B}(\mathbb{R}^{m+1})$, that is, $\lambda_{m+1}$ is the image measure of $\lambda_{m}\otimes\lambda_{1}$ under the measurable map $\iota^{m,1}$, and claim 2 of Image Measures, Measures with Densities, and Change of Variables gives (2) for $n=1$.

Step 3 (Rebracketing). Let $m,n\in\mathbb{N}$, $u\in\mathbb{R}^{m}$, $w\in\mathbb{R}^{n}$ and $t\in\mathbb{R}^{1}$. By claim 3 of Arithmetic of Addition on the Natural Numbers, $m+(n+1)=(m+n)+1$, and we claim

By the description of indices in Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space, every $k\in[m+(n+1)]$ satisfies exactly one of $k\in[m]$ and $k=m+j$ with a unique $j\in[n+1]$, and by claim 5 of Properties of the Order on the Natural Numbers such a $j$ satisfies $j\in[n]$ or $j=n+1$. If $k\in[m]$, then $k\in[m+n]$ by claim 6 of Properties of the Order on the Natural Numbers, and both sides of (3) have $k$th coordinate $u_{k}$. If $k=m+j$ with $j\in[n]$, the left side has $k$th coordinate equal to the $j$th coordinate of $\iota^{n,1}(w,t)$, namely $w_{j}$; and $k\in[m+n]$ by claim 6 of Properties of the Order on the Natural Numbers, so the right side has $k$th coordinate equal to the $k$th coordinate of $\iota^{m,n}(u,w)$, again $w_{j}$. If $k=m+(n+1)=(m+n)+1$, both sides have $k$th coordinate $t$. This proves (3).

Step 4 ($n\in S$ implies $n+1\in S$). Let $n\in S$, let $m\in\mathbb{N}$ and let $F:\mathbb{R}^{m+(n+1)}\to[0,\infty]$ be Borel. For $u\in\mathbb{R}^{m}$ let $g_{u}(w)=F(\iota^{m,n+1}(u,w))$ for $w\in\mathbb{R}^{n+1}$ and $K(u)=\int_{\mathbb{R}^{n+1}}g_{u}\,d\lambda_{n+1}$; by Step 1 (with $n+1$ in place of $n$) each $g_{u}$ is Borel and

Define $G:\mathbb{R}^{m+n}\to[0,\infty]$ by $G(z)=\int_{\mathbb{R}^{1}}F(\iota^{m+n,1}(z,t))\,\lambda_{1}(dt)$, which is Borel by Step 1 applied to $F$ on $\mathbb{R}^{(m+n)+1}=\mathbb{R}^{m+(n+1)}$. By $1\in S$ (Step 2) applied with $n$ in place of $m$ to $g_{u}$, then Step 1 with $(n,1)$ in place of $(m,n)$, and then (3),

On the other hand, by $1\in S$ applied with $m+n$ in place of $m$, Step 1 with $(m+n,1)$, then $n\in S$ applied to $G$, and Step 1 with $(m,n)$,

The inner integral on the right is $K(u)$, so by (4) the equality (2) holds for $m$ and $n+1$. As $m$ and $F$ were arbitrary, $n+1\in S$.

Step 5 (Conclusion). By Steps 1 to 4, $S=\mathbb{N}$, so (2) holds for all $m,n\in\mathbb{N}$; as noted in Step 1, (1) and (2) with $m=q$, $n=p$ and $F=f$ give the Borel measurability of $v\mapsto f(\iota^{q,p}(u,v))$ for each $u$, the Borel measurability of $u\mapsto\int_{\mathbb{R}^{p}}f(\iota^{q,p}(u,v))\,\lambda_{p}(dv)$, and the displayed identity of the lemma.