The Borel σ-Algebra of [0,∞]

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $[0,\infty]$ and $<$ be as in The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (iii), and let $\mathcal{P}([0,\infty])$ be the power set of $[0,\infty]$.

(i) For $a\in\mathbb{R}$, the ray $(a,\infty]$ is the subset

$$(a,\infty]=\{x\in[0,\infty]:a<x\}$$

of $[0,\infty]$, and $\mathcal{R}$ is the set of those $A\in\mathcal{P}([0,\infty])$ for which there is $a\in\mathbb{R}$ with $A=(a,\infty]$. So $\mathcal{R}$ is a set of subsets of $[0,\infty]$, and its elements are exactly the rays $(a,\infty]$ with $a\in\mathbb{R}$.

(ii) The Borel $\sigma$-algebra of $[0,\infty]$ is the $\sigma$-algebra on $[0,\infty]$ generated by $\mathcal{R}$,

$$\mathcal{B}([0,\infty])=\sigma(\mathcal{R}).$$

Its members are the Borel subsets of $[0,\infty]$.