A Closed Subset of a Euclidean Space is a Countable Union of Compact Sets, and Its Coordinate Projections are Borel

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, whose Euclidean spaces, norm, distance $d_{E}$, topology and closed balls $\bar{B}(x,r)$ are those fixed there, let $m,q,p\in\mathbb{N}$ satisfy $1\le m$, $1\le q$ and $1\le p$. The natural number $k$ is read as a real number through the canonical map $\iota$ from $\mathbb{N}$ to $\mathbb{R}$ where a real number is required; this is the only use of the symbol $\iota$ here, the concatenation map being written $\iota^{q,p}$ throughout. Compact is compactness in a metric space and closed is closedness for the topology of $d_{E}$.

1. (Exhaustion by compact sets)(i) Let $C\subseteq\mathbb{R}^{m}$ be closed. For $k\in\mathbb{N}$ the set $C_{k}=C\cap\bar{B}(0_{\mathbb{R}^{m}},\iota(k))$ is compact, being closed and bounded and hence compact by The Heine-Borel Theorem in $\mathbb{R}^n$, and Compactness of Closed Balls and the Unit Sphere, and

$$C=\bigcup_{k\in\mathbb{N}}C_{k}.$$

2. (Projections of a closed set are Borel)(ii) Let $C\subseteq\mathbb{R}^{q+p}$ be closed, and let $\mathrm{pr}^{q,p}_{1}$ and $\mathrm{pr}^{q,p}_{2}$ be the coordinate projections. Then

$$\mathrm{pr}^{q,p}_{1}(C)=\{\mathrm{pr}^{q,p}_{1}(z):z\in C\}\in\mathcal{B}(\mathbb{R}^{q}),\qquad \mathrm{pr}^{q,p}_{2}(C)=\{\mathrm{pr}^{q,p}_{2}(z):z\in C\}\in\mathcal{B}(\mathbb{R}^{p}).$$