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

Let $l\ge1$ be a natural number. Write $\mathcal{B}(\mathbb{R})$ for the Borel $\sigma$-algebra on the real numbers and $\lambda$ for Lebesgue measure on it. Members of the $l$-fold Cartesian product $\mathbb{R}^{l}$ are written as tuples $\theta=(\theta_1,\dots,\theta_l)$, and for $l\ge2$ we identify $\mathbb{R}^{l}$ with $\mathbb{R}^{l-1}\times\mathbb{R}$ by identifying $(\theta_1,\dots,\theta_l)$ with $((\theta_1,\dots,\theta_{l-1}),\theta_l)$. Define $\mathcal{B}_1=\mathcal{B}(\mathbb{R})$ and $\lambda_1=\lambda$, and recursively for $l\ge2$ the product $\sigma$-algebra $\mathcal{B}_l=\mathcal{B}_{l-1}\otimes\mathcal{B}(\mathbb{R})$ and the product measure $\lambda_l=\lambda_{l-1}\otimes\lambda$ on it; under the identification, $\mathcal{B}_l$ is regarded as a $\sigma$-algebra of subsets of $\mathbb{R}^{l}$ and $\lambda_l$ as a measure on it (the identification is a bijection, along which sets, $\sigma$-algebras, and measures are transported elementwise). A Borel rectangle in $\mathbb{R}^{l}$ is a set $A_1\times\dots\times A_l$ with every $A_j\in\mathcal{B}(\mathbb{R})$, read as $A_1$ when $l=1$. Products in $[0,\infty]$ use the conventions of Measure, Measure Space, and Probability Measure, extended by $a\cdot\infty=\infty\cdot a=\infty$ for $0<a\le\infty$. For $l=1$ we further use, in claims 2, 3, and 4, the conventions that $\mathbb{R}^{l-1}\times Y$ denotes $Y$, that $\mathcal{B}_{l-1}\otimes\mathcal{G}$ denotes $\mathcal{G}$, that $\lambda_{l-1}\otimes\mu$ denotes $\mu$, that a pair $(\theta',y)$ denotes just $y$, and that the insertion map below is the identity presentation $\Psi_1(t,y)=(t,y)$.

1. (Generation, rectangle values, uniqueness) $\mathcal{B}_l$ is generated by the Borel rectangles, and $\lambda_l$ is a $\sigma$-finite measure on $\mathcal{B}_l$ with

for every Borel rectangle, the product taken in $[0,\infty]$ with the conventions above; moreover $\lambda_l$ is the only measure on $\mathcal{B}_l$ with these rectangle values.

2. (Coordinate insertion) Suppose $l\ge1$, let $(Y,\mathcal{G},\mu)$ be a measure space whose measure $\mu$ is $\sigma$-finite, and let $i\in\{1,\dots,l\}$. Then $\mathcal{B}_l\otimes\mathcal{G}$ is generated by the sets $R\times G$ with $R$ a Borel rectangle in $\mathbb{R}^{l}$ and $G\in\mathcal{G}$. The insertion map

is a bijection; it is measurable with measurable inverse, with respect to $\mathcal{B}(\mathbb{R})\otimes(\mathcal{B}_{l-1}\otimes\mathcal{G})$ and $\mathcal{B}_l\otimes\mathcal{G}$; and the image measure of $\lambda\otimes(\lambda_{l-1}\otimes\mu)$ under $\Psi_i$ is $\lambda_l\otimes\mu$, all products of $\sigma$-finite measures here being formed with Existence and Uniqueness of the Product Measure.

3. (Coordinate Tonelli) In the setting of claim 2, let $f:\mathbb{R}^{l}\times Y\to[0,\infty]$ be measurable with respect to $\mathcal{B}_l\otimes\mathcal{G}$, in the sense of Lebesgue Integral of a Nonnegative Measurable Function. Then for every $(\theta',y)\in\mathbb{R}^{l-1}\times Y$ the map $t\mapsto f\bigl(\Psi_i(t,(\theta',y))\bigr)$ is measurable with respect to $\mathcal{B}(\mathbb{R})$; the map $(\theta',y)\mapsto\int_{\mathbb{R}}f\bigl(\Psi_i(t,(\theta',y))\bigr)\,d\lambda(t)$ is measurable with respect to $\mathcal{B}_{l-1}\otimes\mathcal{G}$; and

4. (Coordinate Fubini) In the setting of claim 2, let $f:\mathbb{R}^{l}\times Y\to\mathbb{R}$ be integrable with respect to $\lambda_l\otimes\mu$. Then there is $N\in\mathcal{B}_{l-1}\otimes\mathcal{G}$ with $(\lambda_{l-1}\otimes\mu)(N)=0$ such that for every $(\theta',y)\notin N$ the map $t\mapsto f\bigl(\Psi_i(t,(\theta',y))\bigr)$ is integrable with respect to $\lambda$; the function equal to $\int_{\mathbb{R}}f\bigl(\Psi_i(t,(\theta',y))\bigr)\,d\lambda(t)$ off $N$ and to $0$ on $N$ is integrable with respect to $\lambda_{l-1}\otimes\mu$; and its integral equals $\int_{\mathbb{R}^{l}\times Y}f\,d(\lambda_l\otimes\mu)$.