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

In the setting of Measurable Functions and the Lebesgue Integral: Notation, let $m,n\in\mathbb{N}$. Let $\iota:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{m+n}$ be the concatenation function, a bijection by Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (iii), and let $\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n})$ be the product $\sigma$-algebra on $\mathbb{R}^{m}\times\mathbb{R}^{n}$ of the Borel $\sigma$-algebras of $\mathbb{R}^{m}$ and $\mathbb{R}^{n}$.

(i) Concatenation is measurable. $\iota$ is $(\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n}))/\mathcal{B}(\mathbb{R}^{m+n})$-measurable.

(ii) Its inverse is measurable. The inverse $\iota^{-1}:\mathbb{R}^{m+n}\to\mathbb{R}^{m}\times\mathbb{R}^{n}$ is $\mathcal{B}(\mathbb{R}^{m+n})/(\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n}))$-measurable. Consequently, for every $E\subseteq\mathbb{R}^{m}\times\mathbb{R}^{n}$,

$$E\in\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}^{n})\iff\iota[E]\in\mathcal{B}(\mathbb{R}^{m+n}).$$

(iii) The real line and $\mathbb{R}^{1}$. For $t\in\mathbb{R}$, let $\tau(t)\in\mathbb{R}^{1}$ be the $1$-tuple with component $\tau(t)_{1}=t$, that is, the constant function on $[1]$ with value $t$. The function $\tau:\mathbb{R}\to\mathbb{R}^{1}$ is a bijection whose inverse is $x\mapsto x_{1}$; $\tau$ is $\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R}^{1})$-measurable, and its inverse is $\mathcal{B}(\mathbb{R}^{1})/\mathcal{B}(\mathbb{R})$-measurable. Consequently, for every $B\subseteq\mathbb{R}$,

$$B\in\mathcal{B}(\mathbb{R})\iff\tau[B]\in\mathcal{B}(\mathbb{R}^{1}).$$