The Borel σ-Algebra of the Real Line Is Generated by the Closed Rays, and by the Open Left Rays

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $\mathcal{B}(\mathbb{R})$ be the Borel $\sigma$-algebra of $\mathbb{R}$, as in Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation (v), and for a set $\mathcal{C}$ of subsets of $\mathbb{R}$ let $\sigma(\mathcal{C})$ be the $\sigma$-algebra on $\mathbb{R}$ generated by $\mathcal{C}$. For $a\in\mathbb{R}$, $[a,\infty)$ and $(-\infty,a)$ are the rays of Intervals of the Real Line (iii), and $\{[a,\infty):a\in\mathbb{R}\}$ and $\{(-\infty,a):a\in\mathbb{R}\}$ are the sets of subsets of $\mathbb{R}$ of The Families of Rays of the Real Line Are Sets of Subsets of the Real Line (i).

(i) $\mathcal{B}(\mathbb{R})=\sigma(\{[a,\infty):a\in\mathbb{R}\})$.

(ii) $\mathcal{B}(\mathbb{R})=\sigma(\{(-\infty,a):a\in\mathbb{R}\})$.