Proof of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$

Throughout, integrals of $[0,\infty]$-valued measurable functions are those of Lebesgue Integral of a Nonnegative Measurable Function, with the $[0,\infty]$ conventions of the statement; $\pi$-systems and $\lambda$-systems are those of Dynkin's Pi-Lambda Theorem; generated $\sigma$-algebras are those of Generated Sigma-Algebra; and $\sigma$-algebra manipulations use Sigma-Algebra and Measurable Space. Write $\mathcal{R}_l$ for the family of Borel rectangles in $\mathbb{R}^{l}$. We use two elementary consequences of countable additivity (Measure, Measure Space, and Probability Measure) for a measure $m$: finite additivity and monotonicity — padding a finite disjoint family with copies of $\varnothing$ gives $m(U\cup V)=m(U)+m(V)$ for disjoint $U,V$, whence $m(U)\le m(V)$ for $U\subseteq V$, as $m(V)=m(U)+m(V\setminus U)$ — and continuity from below: if $E_1\subseteq E_2\subseteq\dots$, then $m(\bigcup_jE_j)=\lim_jm(E_j)$, since the union is the disjoint union of the successive differences $E_j\setminus E_{j-1}$ (with $E_0=\varnothing$) and the partial sums of their masses are the $m(E_j)$.

Step 1 (a uniqueness principle). We first record: let $Z$ be a set, $\mathcal{P}$ a $\pi$-system of subsets of $Z$, and $m$, $m'$ measures on $\sigma(\mathcal{P})$ with $m(C)=m'(C)$ for every $C\in\mathcal{P}$; suppose there are pairwise disjoint $Z_1,Z_2,\dots\in\mathcal{P}$ with $\bigcup_nZ_n=Z$ and $m(Z_n)<\infty$ for every $n$. Then $m=m'$.

Indeed, fix $n$ and let $\mathcal{D}_n$ be the class of $E\in\sigma(\mathcal{P})$ with $m(E\cap Z_n)=m'(E\cap Z_n)$. It contains $Z$, both sides being the common finite value $m(Z_n)=m'(Z_n)$ (as $Z_n\in\mathcal{P}$). It is closed under proper differences: for $F\subseteq E$ both in $\mathcal{D}_n$, finite additivity gives $m(E\cap Z_n)=m((E\setminus F)\cap Z_n)+m(F\cap Z_n)$, all terms being finite by monotonicity (each is at most $m(Z_n)<\infty$), so $m((E\setminus F)\cap Z_n)=m(E\cap Z_n)-m(F\cap Z_n)$, and likewise for $m'$. It is closed under increasing countable unions by continuity from below. Hence $\mathcal{D}_n$ is a $\lambda$-system; it contains $\mathcal{P}$, since for $E\in\mathcal{P}$ the set $E\cap Z_n$ again lies in the $\pi$-system $\mathcal{P}$, on which $m$ and $m'$ agree; so $\mathcal{D}_n\supseteq\sigma(\mathcal{P})$ by Dynkin's Pi-Lambda Theorem. Finally, for $E\in\sigma(\mathcal{P})$, countable additivity along the pairwise disjoint sets $E\cap Z_n$ gives $m(E)=\sum_nm(E\cap Z_n)=\sum_nm'(E\cap Z_n)=m'(E)$.

Step 2 (claim 1). Induction on $l$. For $l=1$ the Borel rectangles are exactly the Borel sets, so generation is trivial; $\lambda$ is $\sigma$-finite by Existence of Lebesgue Measure on the Real Line; and the rectangle values prescribe the measure on every Borel set, so uniqueness is immediate. Let $l\ge2$ and assume claim 1 for $l-1$.

Generation. Every Borel rectangle $A_1\times\dots\times A_l=(A_1\times\dots\times A_{l-1})\times A_l$ is a measurable rectangle of $\mathcal{B}_{l-1}\otimes\mathcal{B}(\mathbb{R})$ (Product Sigma-Algebra), since $A_1\times\dots\times A_{l-1}\in\mathcal{B}_{l-1}$ by the induction hypothesis; hence $\sigma(\mathcal{R}_l)\subseteq\mathcal{B}_l$. Conversely, fix $A\in\mathcal{B}(\mathbb{R})$ and let $\mathcal{C}_A$ be the class of all $H\subseteq\mathbb{R}^{l-1}$ with $H\times A\in\sigma(\mathcal{R}_l)$. Then $\mathbb{R}^{l-1}\in\mathcal{C}_A$, as $\mathbb{R}^{l-1}\times A\in\mathcal{R}_l$; $\mathcal{C}_A$ is closed under complements, since $(\mathbb{R}^{l-1}\setminus H)\times A=(\mathbb{R}^{l-1}\times A)\setminus(H\times A)$ and $\sigma$-algebras are closed under set differences; and it is closed under countable unions, since $(\bigcup_mH_m)\times A=\bigcup_m(H_m\times A)$. So $\mathcal{C}_A$ is a $\sigma$-algebra containing $\mathcal{R}_{l-1}$, hence $\mathcal{C}_A\supseteq\mathcal{B}_{l-1}$ by the induction hypothesis. Thus every measurable rectangle $H\times A$ generating $\mathcal{B}_{l-1}\otimes\mathcal{B}(\mathbb{R})$ lies in $\sigma(\mathcal{R}_l)$, and $\mathcal{B}_l\subseteq\sigma(\mathcal{R}_l)$.

Rectangle values and $\sigma$-finiteness. By Existence and Uniqueness of the Product Measure, $\lambda_l=\lambda_{l-1}\otimes\lambda$ is a $\sigma$-finite measure on $\mathcal{B}_l$ with $\lambda_l((A_1\times\dots\times A_{l-1})\times A_l)=\lambda_{l-1}(A_1\times\dots\times A_{l-1})\,\lambda(A_l)$, which equals $\lambda(A_1)\cdots\lambda(A_l)$ by the induction hypothesis.

Uniqueness. $\mathcal{R}_l$ is a $\pi$-system, intersections being formed coordinatewise: $(A_1\times\dots\times A_l)\cap(A_1'\times\dots\times A_l')=(A_1\cap A_1')\times\dots\times(A_l\cap A_l')$. For each $l$-tuple $q=(q_1,\dots,q_l)$ of integers put $Z_q=[q_1,q_1+1)\times\dots\times[q_l,q_l+1)$; each factor is an interval, hence a Borel set of Lebesgue measure $1$ by Existence of Lebesgue Measure on the Real Line, so the $Z_q$ are countably many pairwise disjoint members of $\mathcal{R}_l$ covering $\mathbb{R}^{l}$ with $\lambda_l(Z_q)=1$ by the rectangle values. Any measure on $\mathcal{B}_l$ with the stated rectangle values agrees with $\lambda_l$ on $\mathcal{R}_l$, so Step 1 gives uniqueness.

Step 3 (claim 2 for $l\ge2$: generation and bimeasurability). Let $l\ge2$, $(Y,\mathcal{G},\mu)$ and $i$ be as in claim 2, and write $\mathcal{P}_{l'}$ for the family of sets $R\times G$ with $R\in\mathcal{R}_{l'}$ and $G\in\mathcal{G}$.

(a) For every $l'\ge1$, $\mathcal{B}_{l'}\otimes\mathcal{G}$ is generated by $\mathcal{P}_{l'}$: each $R\times G$ is a measurable rectangle of $\mathcal{B}_{l'}\otimes\mathcal{G}$, since $R\in\sigma(\mathcal{R}_{l'})=\mathcal{B}_{l'}$ by claim 1; conversely, for fixed $G\in\mathcal{G}$, the class of $H\subseteq\mathbb{R}^{l'}$ with $H\times G\in\sigma(\mathcal{P}_{l'})$ is a $\sigma$-algebra (it contains $\mathbb{R}^{l'}$ as $\mathbb{R}^{l'}\times G\in\mathcal{P}_{l'}$, and is closed under complements and countable unions exactly as in Step 2) containing $\mathcal{R}_{l'}$, hence containing $\mathcal{B}_{l'}$ by claim 1; so every generator $H\times G$ of $\mathcal{B}_{l'}\otimes\mathcal{G}$ lies in $\sigma(\mathcal{P}_{l'})$. This proves in particular the generation assertion of claim 2.

(b) $\mathcal{B}(\mathbb{R})\otimes(\mathcal{B}_{l-1}\otimes\mathcal{G})$ is generated by the family $\mathcal{Q}$ of sets $A\times(R'\times G)$ with $A\in\mathcal{B}(\mathbb{R})$, $R'\in\mathcal{R}_{l-1}$, $G\in\mathcal{G}$: each such set is a measurable rectangle, since $R'\times G\in\mathcal{B}_{l-1}\otimes\mathcal{G}$; conversely, for fixed $A\in\mathcal{B}(\mathbb{R})$, the class of $K\subseteq\mathbb{R}^{l-1}\times Y$ with $A\times K\in\sigma(\mathcal{Q})$ is a $\sigma$-algebra (it contains $\mathbb{R}^{l-1}\times Y$ since $A\times(\mathbb{R}^{l-1}\times Y)\in\mathcal{Q}$, and is closed under complements and countable unions as before) containing the sets $R'\times G$, hence containing $\mathcal{B}_{l-1}\otimes\mathcal{G}$ by (a) applied with $l'=l-1$; so every generator $A\times K$ lies in $\sigma(\mathcal{Q})$.

$\Psi_i$ is a bijection; its inverse $\Xi$ sends $(\theta,y)$ to $(\theta_i,((\theta_1,\dots,\theta_{i-1},\theta_{i+1},\dots,\theta_l),y))$. For $R=A_1\times\dots\times A_l\in\mathcal{R}_l$ and $G\in\mathcal{G}$,

and for $A\in\mathcal{B}(\mathbb{R})$, $R'=A_1'\times\dots\times A_{l-1}'\in\mathcal{R}_{l-1}$, $G\in\mathcal{G}$,

By the generator criterion of Measurable Function and Real-Valued Measurable Function together with (a) and (b), both $\Psi_i$ and $\Xi$ are measurable.

Step 4 (claim 2 for $l\ge2$: the image measure). The measures $\lambda_{l-1}\otimes\mu$, $\lambda_l\otimes\mu$ and $\lambda\otimes(\lambda_{l-1}\otimes\mu)$ are defined and $\sigma$-finite by Existence and Uniqueness of the Product Measure, their factors being $\sigma$-finite by claim 1 and by hypothesis. Let $m$ be the image measure of $\lambda\otimes(\lambda_{l-1}\otimes\mu)$ under $\Psi_i$, a measure on $\mathcal{B}_l\otimes\mathcal{G}$ by Step 3 and claim 1 of Image Measures, Measures with Densities, and Change of Variables. For $R=A_1\times\dots\times A_l\in\mathcal{R}_l$ and $G\in\mathcal{G}$, the preimage computed in Step 3 and the rectangle property of Existence and Uniqueness of the Product Measure, applied twice, give

using the rectangle values of claim 1 for $\lambda_{l-1}$ and for $\lambda_l$. The family $\mathcal{P}_l$ is a $\pi$-system (coordinatewise intersections). By $\sigma$-finiteness of $\mu$ there are $Y_1'',Y_2'',\dots\in\mathcal{G}$ covering $Y$ with $\mu(Y_n'')<\infty$; setting $Y_1=Y_1''$ and $Y_n=Y_n''\setminus(Y_1''\cup\dots\cup Y_{n-1}'')$ yields pairwise disjoint members of $\mathcal{G}$ with $\bigcup_nY_n=Y$ and $\mu(Y_n)\le\mu(Y_n'')<\infty$ by monotonicity. The countably many sets $Z_q\times Y_n$ (with $Z_q$ as in Step 2) are pairwise disjoint members of $\mathcal{P}_l$ covering $\mathbb{R}^{l}\times Y$, with $(\lambda_l\otimes\mu)(Z_q\times Y_n)=\mu(Y_n)<\infty$. Step 1 applied to $m$ and $\lambda_l\otimes\mu$ gives $m=\lambda_l\otimes\mu$.

Step 5 (claims 3 and 4 for $l\ge2$). Write $m_0=\lambda\otimes(\lambda_{l-1}\otimes\mu)$. Let $f:\mathbb{R}^{l}\times Y\to[0,\infty]$ be $\mathcal{B}_l\otimes\mathcal{G}$-measurable. Then $f\circ\Psi_i$ is measurable, since for real $a$ one has $\{f\circ\Psi_i>a\}=\Psi_i^{-1}(\{f>a\})$, measurable by Step 3; and claim 2 of Image Measures, Measures with Densities, and Change of Variables, applied to $T=\Psi_i$ whose image measure is $\lambda_l\otimes\mu$ by Step 4, gives

The Sections clause of Tonelli and Fubini Theorems, applied to the $\sigma$-finite factors $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$ and $(\mathbb{R}^{l-1}\times Y,\mathcal{B}_{l-1}\otimes\mathcal{G},\lambda_{l-1}\otimes\mu)$, shows that for every $(\theta',y)$ the section $t\mapsto(f\circ\Psi_i)(t,(\theta',y))=f(\Psi_i(t,(\theta',y)))$ is $\mathcal{B}(\mathbb{R})$-measurable, and the Tonelli clause — the second iterated integral of its display, which integrates over the first factor innermost — shows that $(\theta',y)\mapsto\int_{\mathbb{R}}f(\Psi_i(t,(\theta',y)))\,d\lambda(t)$ is $\mathcal{B}_{l-1}\otimes\mathcal{G}$-measurable with

Chaining the two displays proves claim 3. For claim 4, let $f:\mathbb{R}^{l}\times Y\to\mathbb{R}$ be integrable with respect to $\lambda_l\otimes\mu$; by the integrable clause of claim 2 of Image Measures, Measures with Densities, and Change of Variables, $f\circ\Psi_i$ is integrable with respect to $m_0$ with the same integral, and the Fubini clause of Tonelli and Fubini Theorems — its symmetric statement in the other order, whose exceptional set lies in the second factor — provides exactly the set $N\in\mathcal{B}_{l-1}\otimes\mathcal{G}$ of $(\lambda_{l-1}\otimes\mu)$-measure zero and the integrability statements and identity asserted in claim 4.

Step 6 (claims 2, 3, and 4 for $l=1$). Under the conventions of the statement: the generation assertion of claim 2 is that $\mathcal{B}(\mathbb{R})\otimes\mathcal{G}$ is generated by the measurable rectangles $A\times G$, which is Product Sigma-Algebra directly; $\Psi_1$ is the identity map of $\mathbb{R}\times Y$, trivially a bijection that is measurable with measurable inverse, and the image measure of $\lambda\otimes\mu$ under it is $\lambda\otimes\mu=\lambda_1\otimes\mu$ itself (its value on each $B\in\mathcal{B}(\mathbb{R})\otimes\mathcal{G}$ being unchanged); and claims 3 and 4 are precisely the Sections, Tonelli, and symmetric Fubini clauses of Tonelli and Fubini Theorems applied to the $\sigma$-finite factors $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$ and $(Y,\mathcal{G},\mu)$, in the order that integrates over $\mathbb{R}$ innermost. $\blacksquare$