Proof of Borel Sets of ℝ^{m+n} and the Product σ-Algebra of ℝ^m and ℝ^n

Throughout, for $p\in\mathbb{N}$ the pair $(\mathbb{R}^{p},\mathcal{B}(\mathbb{R}^{p}))$ is a measurable space and a function into $\mathbb{R}^{p}$ is measurable when it is measurable for $\mathcal{B}(\mathbb{R}^{p})$ (Measurable Map, and the Set of Measurable Maps (iv)); the criterion used for all three clauses is

$$u:W\to\mathbb{R}^{p}\text{ is measurable}\iff\text{each component function }w\mapsto u(w)_{k}\ (k\in[p])\text{ is measurable},$$

for every measurable space $(W,\mathcal{K})$, by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i) (components as in Component Maps and Entry Maps (i)). For $k\in[p]$ let $\pi_{k}:\mathbb{R}^{p}\to\mathbb{R}$, $\pi_{k}(z)=z_{k}$, be the $k$-th component function of the identity of $\mathbb{R}^{p}$. The identity is $\mathcal{B}(\mathbb{R}^{p})/\mathcal{B}(\mathbb{R}^{p})$-measurable (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iii)), so by the criterion every coordinate function $\pi_{k}$ is $\mathcal{B}(\mathbb{R}^{p})/\mathcal{B}(\mathbb{R})$-measurable. Write $\mathcal{P}=\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n})$, and let $\mathrm{pr}_{1}:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{m}$ and $\mathrm{pr}_{2}:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the projections, which are $\mathcal{P}/\mathcal{B}(\mathbb{R}^{m})$- and $\mathcal{P}/\mathcal{B}(\mathbb{R}^{n})$-measurable by The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (vi).

Concatenation. Let $k\in[m+n]$. By Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (i), either $k\in[m]$, or $k=m+j$ for some $j\in[n]$. By Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (ii), in the first case

$$\iota(\xi,\eta)_{k}=\xi_{k}=\pi_{k}\bigl(\mathrm{pr}_{1}(\xi,\eta)\bigr),$$

so the $k$-th component function of $\iota$ is $\pi_{k}\circ\mathrm{pr}_{1}$; in the second case

$$\iota(\xi,\eta)_{m+j}=\eta_{j}=\pi_{j}\bigl(\mathrm{pr}_{2}(\xi,\eta)\bigr),$$

so it is $\pi_{j}\circ\mathrm{pr}_{2}$. Either way it is $\mathcal{P}/\mathcal{B}(\mathbb{R})$-measurable as a composite of measurable functions (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (ii)). By the criterion, $\iota$ is $\mathcal{P}/\mathcal{B}(\mathbb{R}^{m+n})$-measurable. This proves (i).

The inverse. Write $\kappa=\iota^{-1}$, and let $a=\mathrm{pr}_{1}\circ\kappa:\mathbb{R}^{m+n}\to\mathbb{R}^{m}$ and $b=\mathrm{pr}_{2}\circ\kappa:\mathbb{R}^{m+n}\to\mathbb{R}^{n}$, so that $\kappa(w)=(a(w),b(w))$ for $w\in\mathbb{R}^{m+n}$. By Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (iii),

$$a(w)_{k}=w_{k}=\pi_{k}(w)\quad(k\in[m]),\qquad b(w)_{j}=w_{m+j}=\pi_{m+j}(w)\quad(j\in[n]),$$

with $\pi_{k}$ and $\pi_{m+j}$ the coordinate functions of $\mathbb{R}^{m+n}$. So the component functions of $a$ and of $b$ are coordinate functions, which are measurable, and by the criterion $a$ is $\mathcal{B}(\mathbb{R}^{m+n})/\mathcal{B}(\mathbb{R}^{m})$-measurable and $b$ is $\mathcal{B}(\mathbb{R}^{m+n})/\mathcal{B}(\mathbb{R}^{n})$-measurable. By The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (vii), $\kappa$ is $\mathcal{B}(\mathbb{R}^{m+n})/\mathcal{P}$-measurable.

Now let $E\subseteq\mathbb{R}^{m}\times\mathbb{R}^{n}$. Then

$$\iota[E]=\kappa^{-1}(E),\qquad E=\iota^{-1}\bigl(\iota[E]\bigr),$$

where $\iota^{-1}(\cdot)$ and $\kappa^{-1}(\cdot)$ denote preimages (Images and Preimages under a Bijection and Its Inverse (i) and Images and Preimages under a Bijection and Its Inverse (ii), applied to the bijection $\iota$ with inverse $\kappa$). If $E\in\mathcal{P}$, then $\iota[E]=\kappa^{-1}(E)\in\mathcal{B}(\mathbb{R}^{m+n})$ by measurability of $\kappa$; if $\iota[E]\in\mathcal{B}(\mathbb{R}^{m+n})$, then $E=\iota^{-1}(\iota[E])\in\mathcal{P}$ by (i). This proves (ii).

The real line and $\mathbb{R}^{1}$. Here $[1]=\{1\}$ (Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (i)), so an element of $\mathbb{R}^{1}$ is determined by its first component (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iii)). The function $\tau$ exists as in Sets and Maps: Ordinary Notation (v), and for $t\in\mathbb{R}$ and $x\in\mathbb{R}^{1}$,

$$\pi_{1}(\tau(t))=\tau(t)_{1}=t,\qquad \tau(\pi_{1}(x))=x,$$

the second because both sides are $1$-tuples with first component $x_{1}$. So $\tau$ is a bijection with inverse $\pi_{1}:x\mapsto x_{1}$, by Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (vii). The coordinate function $\pi_{1}$ is $\mathcal{B}(\mathbb{R}^{1})/\mathcal{B}(\mathbb{R})$-measurable, as shown above. The only component function of $\tau$ is $\pi_{1}\circ\tau=\mathrm{id}_{\mathbb{R}}$, which is $\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R})$-measurable (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iii)); by the criterion, $\tau$ is $\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R}^{1})$-measurable.

Finally let $B\subseteq\mathbb{R}$. By Images and Preimages under a Bijection and Its Inverse (i) and Images and Preimages under a Bijection and Its Inverse (ii), applied to the bijection $\tau$ with inverse $\pi_{1}$,

$$\tau[B]=\pi_{1}^{-1}(B),\qquad B=\tau^{-1}\bigl(\tau[B]\bigr).$$

If $B\in\mathcal{B}(\mathbb{R})$, then $\tau[B]=\pi_{1}^{-1}(B)\in\mathcal{B}(\mathbb{R}^{1})$ by measurability of $\pi_{1}$; if $\tau[B]\in\mathcal{B}(\mathbb{R}^{1})$, then $B=\tau^{-1}(\tau[B])\in\mathcal{B}(\mathbb{R})$ by measurability of $\tau$. This proves (iii).