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

Let $\mathcal{C}=\{[a,\infty):a\in\mathbb{R}\}$, $\mathcal{L}=\{(-\infty,a):a\in\mathbb{R}\}$ and $\mathcal{R}=\{(a,\infty):a\in\mathbb{R}\}$; these are sets of subsets of $\mathbb{R}$, whose members are exactly the rays of the given form with real endpoint, by The Families of Rays of the Real Line Are Sets of Subsets of the Real Line (i). For each of them, $\sigma(\cdot)$ is a $\sigma$-algebra on $\mathbb{R}$ containing it, by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii). By Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (iii), $\mathcal{B}(\mathbb{R})=\sigma(\mathcal{R})$, so $\mathcal{B}(\mathbb{R})$ is a $\sigma$-algebra on $\mathbb{R}$.

Proof of (i). For every $a\in\mathbb{R}$, $[a,\infty)\in\mathcal{B}(\mathbb{R})$ by Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (ii); so $\mathcal{C}\subseteq\mathcal{B}(\mathbb{R})$, and $\sigma(\mathcal{C})\subseteq\mathcal{B}(\mathbb{R})$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv).

For the converse inclusion, let $a\in\mathbb{R}$, let $\kappa:\mathbb{Q}\to\mathbb{R}$ be the canonical embedding of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (iv), so that $\kappa[\mathbb{Q}]$ is the set of rational numbers read in $\mathbb{R}$, as in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (v), and let $D=\{s\in\kappa[\mathbb{Q}]:a<s\}$. The set $\kappa[\mathbb{Q}]$ is countable by Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (ix) and Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (vi), so its subset $D$ is countable by Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (iv).

By Intervals of the Real Line as Defined Set Symbols, Typed Terms and Values of Maps (iii), with $\mathbb{R}$ for its $D$ and the identity of $\mathbb{R}$ for $s$ and $t$, there is a map from $\mathbb{R}$ to the power set $\mathcal{P}(\mathbb{R})$ with value $[s,\infty)$ at every $s\in\mathbb{R}$. Its restriction to $D$ has the values $[s,\infty)\in\mathcal{C}\subseteq\sigma(\mathcal{C})$, so it is a map $A:D\to\sigma(\mathcal{C})$ with $A_{s}=[s,\infty)$ for every $s\in D$. As $D$ is countable, Unions and Intersections over a Countable Index Set in a σ-Algebra (i), applied to the $\sigma$-algebra $\sigma(\mathcal{C})$ on $\mathbb{R}$, gives

$$\bigcup_{s\in D}[s,\infty)\in\sigma(\mathcal{C}),$$

the union of the family $(A_{s})_{s\in D}$. This union is $(a,\infty)$. Indeed, if $x\in[s,\infty)$ for some $s\in D$, then $a<s\le x$, so $a<x$ and $x\in(a,\infty)$. Conversely, let $x\in(a,\infty)$, that is, $x\in\mathbb{R}$ and $a<x$. By The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (x) there is $u\in\mathbb{Q}$ with $a<\kappa(u)<x$; then $s=\kappa(u)$ lies in $D$ and $s\le x$, so $x\in[s,\infty)=A_{s}$ and $x$ lies in the union, by Indexed Families of Sets and Their Union, Intersection and Product (ii).

Hence $(a,\infty)\in\sigma(\mathcal{C})$ for every $a\in\mathbb{R}$, that is, $\mathcal{R}\subseteq\sigma(\mathcal{C})$, and $\mathcal{B}(\mathbb{R})=\sigma(\mathcal{R})\subseteq\sigma(\mathcal{C})$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (v). Together with the first inclusion, $\mathcal{B}(\mathbb{R})=\sigma(\mathcal{C})$.

Proof of (ii). For every $a\in\mathbb{R}$, $(-\infty,a)=\mathbb{R}\setminus[a,\infty)$ and $[a,\infty)=\mathbb{R}\setminus(-\infty,a)$, as for $x\in\mathbb{R}$, $x<a$ holds if and only if $a\le x$ fails, by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders (vii) applied to the total order $\le$ of $\mathbb{R}$. Since $\sigma(\mathcal{C})$ is a $\sigma$-algebra on $\mathbb{R}$ containing $\mathcal{C}$, $(-\infty,a)=\mathbb{R}\setminus[a,\infty)\in\sigma(\mathcal{C})$ by σ-Algebra and Measurable Space (i); so $\mathcal{L}\subseteq\sigma(\mathcal{C})$ and $\sigma(\mathcal{L})\subseteq\sigma(\mathcal{C})$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (v). Likewise $[a,\infty)=\mathbb{R}\setminus(-\infty,a)\in\sigma(\mathcal{L})$ for every $a\in\mathbb{R}$, so $\mathcal{C}\subseteq\sigma(\mathcal{L})$ and $\sigma(\mathcal{C})\subseteq\sigma(\mathcal{L})$. Hence $\sigma(\mathcal{L})=\sigma(\mathcal{C})=\mathcal{B}(\mathbb{R})$, by (i).