The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets

Let $m$ and $n$ be natural numbers with $m\ge1$ and $n\ge1$. For each natural number $l\ge1$ write $\mathcal{B}_l$ for the $\sigma$-algebra on Euclidean space $\mathbb{R}^l$ built in Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$, which by claim 1 there is generated by the Borel rectangles of $\mathbb{R}^l$, and which for $l=1$ is the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ of the real line. Let $d$ denote the Euclidean distance, a metric on each Euclidean space. A subset of a Euclidean space is called closed when its complement is Euclidean open; by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$ and Metric Open Sets Form a Topology this is the notion of closed set for the metric topology of $d$.

1. (Coordinate projections) For $j\in\{1,\dots,m\}$ the projection $\pi_j:\mathbb{R}^m\to\mathbb{R}$, $\pi_j(x_1,\dots,x_m)=x_j$, is measurable with respect to $\mathcal{B}_m$ and $\mathcal{B}(\mathbb{R})$.

2. (Componentwise criterion) Let $(X,\mathcal{F})$ be a measurable space and let $f=(f^1,\dots,f^m):X\to\mathbb{R}^m$. Then $f$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}_m$ if and only if each component $f^j:X\to\mathbb{R}$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R})$.

3. (Sequentially continuous maps) Call a map $g:\mathbb{R}^n\to\mathbb{R}$ sequentially continuous if $g(x^k)\to g(x)$ whenever $(x^k)_{k\in\mathbb{N}}$ is a sequence in $\mathbb{R}^n$ and $x\in\mathbb{R}^n$ with $d(x^k,x)\to0$, all convergence of real numbers being convergence of real sequences.

(a) Every sequentially continuous $g:\mathbb{R}^n\to\mathbb{R}$ is measurable with respect to $\mathcal{B}_n$ and $\mathcal{B}(\mathbb{R})$. In particular so is every map $g:\mathbb{R}^n\to\mathbb{R}$ that is continuous from $(\mathbb{R}^n,d)$ to $(\mathbb{R},d)$.

(b) If every component of a map $\Phi=(\Phi^1,\dots,\Phi^m):\mathbb{R}^n\to\mathbb{R}^m$ is sequentially continuous, then $\Phi$ is measurable with respect to $\mathcal{B}_n$ and $\mathcal{B}_m$.

4. (Open and closed sets) Every Euclidean open subset of $\mathbb{R}^m$ and every closed subset of $\mathbb{R}^m$ belongs to $\mathcal{B}_m$.

5. (Product description of the Borel $\sigma$-algebra) $\mathcal{B}_m$ is exactly the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^m)$.