Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets

In the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, with Measurable Functions and the Lebesgue Integral: Standing Notation in force for measures and integrals, let $q,p,l\in\mathbb{N}$ satisfy $1\le q$, $1\le p$ and $1\le l$, and let $2=1+1$. Points of a Euclidean space are read as tuples by Euclidean Points as Tuples of Real Numbers, the $k$th component of $x$ being written $x_{k}$. Measurability of maps between measurable spaces is that of Measurable Map, and the Set of Measurable Maps, and $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra of the real line. A map between Euclidean spaces, or from a Euclidean space to $\mathbb{R}$, is called Borel when it is measurable with respect to the Borel $\sigma$-algebras of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background (iv), the target $\mathbb{R}$ carrying $\mathcal{B}(\mathbb{R})$; a map from a Euclidean space $\mathbb{R}^{m}$ to $[0,\infty]$ is called Borel when it is measurable in the sense of Measurable Functions and the Lebesgue Integral: Standing Notation (iii) for the measurable space $(\mathbb{R}^{m},\mathcal{B}(\mathbb{R}^{m}))$, the two readings agreeing for real-valued maps by that clause. By claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{m})$ is the $\sigma$-algebra $\mathcal{B}_{m}$ of that lemma, whose claims are thereby in force for it. Continuity of a map between Euclidean spaces, or into $\mathbb{R}$, is continuity for the Euclidean distances, where by $\mathbb{R}^1$ with the Euclidean Distance Is Isometric to the Real Line the Euclidean distance of $\mathbb{R}$, identified with $\mathbb{R}^{1}$, is the absolute-value metric; a continuous map into $\mathbb{R}$ is Borel by claim 3 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets. $\otimes$ denotes the product $\sigma$-algebra, $\times$ the Cartesian product of sets, and $\iota=\iota^{q,p}:\mathbb{R}^{q}\times\mathbb{R}^{p}\to\mathbb{R}^{q+p}$ the concatenation map, a bijection by claim 1 there: for $x\in\mathbb{R}^{q}$ and $y\in\mathbb{R}^{p}$, the point $\iota(x,y)$ has $i$th component $x_{i}$ for $i\in[q]$ and $y_{j}$ for $i=q+j$ with $j\in[p]$. Then the following hold.

1. (Coordinate projections)(i) There are unique maps $\mathrm{pr}^{q,p}_{1}:\mathbb{R}^{q+p}\to\mathbb{R}^{q}$ and $\mathrm{pr}^{q,p}_{2}:\mathbb{R}^{q+p}\to\mathbb{R}^{p}$ with $\mathrm{pr}^{q,p}_{1}(\iota(x,y))=x$ and $\mathrm{pr}^{q,p}_{2}(\iota(x,y))=y$ for all $x\in\mathbb{R}^{q}$ and $y\in\mathbb{R}^{p}$; every $z\in\mathbb{R}^{q+p}$ equals $\iota(\mathrm{pr}^{q,p}_{1}(z),\mathrm{pr}^{q,p}_{2}(z))$; and both projections are Borel, with $\lVert\mathrm{pr}^{q,p}_{1}(z)\rVert\le\lVert z\rVert$ and $\lVert\mathrm{pr}^{q,p}_{2}(z)\rVert\le\lVert z\rVert$ for every $z\in\mathbb{R}^{q+p}$.

2. (Pairing of maps)(ii) Let $(E,\mathcal{E})$ be a measurable space and let $u:E\to\mathbb{R}^{q}$ and $v:E\to\mathbb{R}^{p}$ be measurable with respect to $\mathcal{E}$ and $\mathcal{B}(\mathbb{R}^{q})$, respectively $\mathcal{B}(\mathbb{R}^{p})$. Then the map $E\to\mathbb{R}^{q+p}$ with value $\iota(u(s),v(s))$ at $s$, denoted $(u,v)$ and called the pairing of $u$ and $v$, is measurable with respect to $\mathcal{E}$ and $\mathcal{B}(\mathbb{R}^{q+p})$. In particular, for Borel $u:\mathbb{R}^{l}\to\mathbb{R}^{q}$ and $v:\mathbb{R}^{l}\to\mathbb{R}^{p}$ the pairing $(u,v):\mathbb{R}^{l}\to\mathbb{R}^{q+p}$ is Borel, and the swap $\sigma^{q,p}:\mathbb{R}^{q+p}\to\mathbb{R}^{p+q}$ with value $\iota^{p,q}(\mathrm{pr}^{q,p}_{2}(z),\mathrm{pr}^{q,p}_{1}(z))$ at $z$ is Borel.

3. (Product $\sigma$-algebra and the product measure on $\mathbb{R}^{q+p}$)(iii) The bijection $\iota$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{q})\otimes\mathcal{B}(\mathbb{R}^{p})$ and $\mathcal{B}(\mathbb{R}^{q+p})$. Consequently, if $\mu$ is a probability measure on $(\mathbb{R}^{q},\mathcal{B}(\mathbb{R}^{q}))$ and $\nu$ is a probability measure on $(\mathbb{R}^{p},\mathcal{B}(\mathbb{R}^{p}))$, then, probability measures being $\sigma$-finite, the product measure $\mu\otimes\nu$ on $\mathcal{B}(\mathbb{R}^{q})\otimes\mathcal{B}(\mathbb{R}^{p})$ has an image measure under $\iota$, denoted $\mu\boxtimes\nu$ and called the product measure on $\mathbb{R}^{q+p}$, which is a probability measure on $(\mathbb{R}^{q+p},\mathcal{B}(\mathbb{R}^{q+p}))$ with the following properties: $(\mu\boxtimes\nu)(\iota(A\times B))=\mu(A)\,\nu(B)$ for all $A\in\mathcal{B}(\mathbb{R}^{q})$ and $B\in\mathcal{B}(\mathbb{R}^{p})$, where $\iota(A\times B)=(\mathrm{pr}^{q,p}_{1})^{-1}(A)\cap(\mathrm{pr}^{q,p}_{2})^{-1}(B)$ belongs to $\mathcal{B}(\mathbb{R}^{q+p})$; the image measures of $\mu\boxtimes\nu$ under $\mathrm{pr}^{q,p}_{1}$ and under $\mathrm{pr}^{q,p}_{2}$ are $\mu$ and $\nu$; and for every Borel $F:\mathbb{R}^{q+p}\to[0,\infty]$ the function $F\circ\iota$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{q})\otimes\mathcal{B}(\mathbb{R}^{p})$, with $\int_{\mathbb{R}^{q+p}}F\,d(\mu\boxtimes\nu)=\int_{\mathbb{R}^{q}\times\mathbb{R}^{p}}F\circ\iota\,d(\mu\otimes\nu)$.

4. (Norm functions)(iv) Write $\lVert x\rVert^{2}=\lVert x\rVert\,\lVert x\rVert$. The maps $x\mapsto\lVert x\rVert$ and $x\mapsto\lVert x\rVert^{2}$ on $\mathbb{R}^{q}$ are Borel; and so are, for $p=q$ and with $\mathrm{pr}_{1}=\mathrm{pr}^{q,q}_{1}$, $\mathrm{pr}_{2}=\mathrm{pr}^{q,q}_{2}$, the maps $z\mapsto\mathrm{pr}_{1}(z)\cdot\mathrm{pr}_{2}(z)$, $z\mapsto\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert$ and $z\mapsto\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert^{2}$ on $\mathbb{R}^{q+q}$. Consequently, for a measurable space $(E,\mathcal{E})$ and measurable $u,v:E\to\mathbb{R}^{q}$, the maps $s\mapsto\lVert u(s)\rVert^{2}$, $s\mapsto u(s)\cdot v(s)$ and $s\mapsto\lVert u(s)-v(s)\rVert^{2}$ are measurable with respect to $\mathcal{E}$ and $\mathcal{B}(\mathbb{R})$. Moreover, for all $x,y\in\mathbb{R}^{q}$,

$$\lVert x-y\rVert^{2}\le2\lVert x\rVert^{2}+2\lVert y\rVert^{2},\qquad \lVert x\rVert^{2}\le2\lVert y\rVert^{2}+2\lVert x-y\rVert^{2},\qquad |x\cdot y|\le\lVert x\rVert\,\lVert y\rVert\le\tfrac12\bigl(\lVert x\rVert^{2}+\lVert y\rVert^{2}\bigr).$$

5. (Finite sets)(v) Every finite subset $F$ of $\mathbb{R}^{q}$ is closed, hence belongs to $\mathcal{B}(\mathbb{R}^{q})$, and so does its complement $\mathbb{R}^{q}\setminus F$; in particular $\{a\}\in\mathcal{B}(\mathbb{R}^{q})$ for every $a\in\mathbb{R}^{q}$. Moreover, for a finite set $F\subseteq\mathbb{R}^{q}$ and a map $w:F\to\mathbb{R}$, the map $\mathbb{R}^{q}\to\mathbb{R}$ equal to $w$ on $F$ and to $0$ off $F$ is Borel.