Borel Subsets of [0,∞]: Rays, the Point ∞, and Traces on ℝ

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $\mathcal{B}([0,\infty])$ be the Borel $\sigma$-algebra of $[0,\infty]$, let $\mathcal{B}(\mathbb{R})$ be the Borel $\sigma$-algebra of $\mathbb{R}$ of Borel Sigma-Algebra of a Metric Space (ii), let $<$, $\le$, $>$ and $\ge$ on $[0,\infty]$ be as in The Extended Real Line, Its Order, and [0,∞] (iii), and let $[0,\infty)=\{x\in\mathbb{R}:0\le x\}$ be the ray of Intervals of the Real Line (iii).

(i) For every $a\in\mathbb{R}$, the sets $\{x\in[0,\infty]:x>a\}$, $\{x\in[0,\infty]:x\ge a\}$, $\{x\in[0,\infty]:x<a\}$ and $\{x\in[0,\infty]:x\le a\}$ belong to $\mathcal{B}([0,\infty])$.

(ii) $\{\infty\}\in\mathcal{B}([0,\infty])$ and $[0,\infty]\setminus\{\infty\}\in\mathcal{B}([0,\infty])$.

(iii) For every subset $B$ of $[0,\infty]$: $B\in\mathcal{B}([0,\infty])$ if and only if $B\cap\mathbb{R}\in\mathcal{B}(\mathbb{R})$.

(iv) Every $C\in\mathcal{B}(\mathbb{R})$ with $C\subseteq[0,\infty)$ belongs to $\mathcal{B}([0,\infty])$.