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

Write $\mathcal{B}=\mathcal{B}([0,\infty])$, and for $a\in\mathbb{R}$ let the ray $(a,\infty]=\{x\in[0,\infty]:a<x\}$ and the set $\mathcal{R}$ of these rays be as in The Borel σ-Algebra of [0,∞] (i), so that $\mathcal{B}=\sigma(\mathcal{R})$ by The Borel σ-Algebra of [0,∞] (ii).

Preliminaries. By The Extended Real Line, Its Order, and [0,∞] (ii), an element $x$ of $[0,\infty]$ is either an element of $\mathbb{R}$ with $0\le x$ or equal to $\infty$, and $\infty\notin\mathbb{R}$ by The Extended Real Line, Its Order, and [0,∞] (i). Hence, with the rays of $\mathbb{R}$ unfolded by Intervals of the Real Line (iii),

$$[0,\infty]\cap\mathbb{R}=[0,\infty),\qquad [0,\infty]=[0,\infty)\cup\{\infty\},\qquad [0,\infty]\setminus\{\infty\}=[0,\infty).$$

For $x,y\in\mathbb{R}$, the relations $x<y$ and $x\le y$ in $[-\infty,\infty]$ hold if and only if they hold in $\mathbb{R}$, by Order and Suprema in the Extended Real Line (i). For $a\in\mathbb{R}$ we have $a\le\infty$, and $a\ne\infty$ because $\infty\notin\mathbb{R}$, so $a<\infty$, by Order and Suprema in the Extended Real Line (ii). In particular, for $a\in\mathbb{R}$, by The Borel σ-Algebra of [0,∞] (i) and Intervals of the Real Line (iii),

$$(a,\infty]\cap\mathbb{R}=\{x\in\mathbb{R}:0\le x,\ a<x\}=(a,\infty)\cap[0,\infty),\qquad (a,\infty]\setminus\{\infty\}=(a,\infty)\cap[0,\infty).$$

By Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), $\mathcal{B}$ is a $\sigma$-algebra on $[0,\infty]$ with $\mathcal{R}\subseteq\mathcal{B}$, so $(a,\infty]\in\mathcal{B}$ for every $a\in\mathbb{R}$ by The Borel σ-Algebra of [0,∞] (i). Likewise $\mathcal{B}(\mathbb{R})$, which is the $\sigma$-algebra on $\mathbb{R}$ generated by the open subsets of $(\mathbb{R},d_{\mathbb{R}})$ by Borel Sigma-Algebra of a Metric Space (i) and Borel Sigma-Algebra of a Metric Space (ii), is a $\sigma$-algebra on $\mathbb{R}$ 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 (ii), $[0,\infty)\in\mathcal{B}(\mathbb{R})$ and $(a,\infty)\in\mathcal{B}(\mathbb{R})$ for every $a\in\mathbb{R}$.

Proof of (ii). The natural numbers are real numbers, by The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (v), so each $k\in\mathbb{N}$ is a real threshold; and, for $k\in\mathbb{N}$, $(k,\infty]=\{y\in[0,\infty]:y>k\}$, as $y>k$ means $k<y$ by The Extended Real Line, Its Order, and [0,∞] (iii), which is the superlevel set at $k$ of the identity map $y\mapsto y$ of $[0,\infty]$, an element of $[0,\infty]^{[0,\infty]}$. So $k\mapsto(k,\infty]$ (Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (vi), Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (v)) is a sequence in $\mathcal{P}([0,\infty])$ whose terms lie in $\mathcal{B}$, hence a sequence in $\mathcal{B}$ (Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (v)), and its intersection $\bigcap_{k\in\mathbb{N}}(k,\infty]$ belongs to $\mathcal{B}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (viii). By Indexed Families of Sets and Their Union, Intersection and Product (iii), its elements are the $x\in[0,\infty]$ with $k<x$ for every $k\in\mathbb{N}$. The point $\infty$ is one of them, since $k<\infty$ for every $k\in\mathbb{N}$ by the preliminaries. If $x\in[0,\infty]$ and $x\ne\infty$, then $x\in\mathbb{R}$, and by The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ix) there is $n\in\mathbb{N}$ with $x<n$; then $n<x$ fails in $\mathbb{R}$, hence in $[-\infty,\infty]$, and $x\notin(n,\infty]$. Therefore $\bigcap_{k\in\mathbb{N}}(k,\infty]=\{\infty\}$, and $\{\infty\}\in\mathcal{B}$. As $\mathcal{B}$ is a $\sigma$-algebra on $[0,\infty]$, $[0,\infty]\setminus\{\infty\}\in\mathcal{B}$ by σ-Algebra and Measurable Space (i); by the preliminaries this set is $[0,\infty)$.

The direction from left to right of (iii). Let $\mathcal{D}$ be the set of those $B\in\mathcal{P}([0,\infty])$, the power set of $[0,\infty]$, with $B\cap\mathbb{R}\in\mathcal{B}(\mathbb{R})$. We check that $\mathcal{D}$ is a $\sigma$-algebra on $[0,\infty]$, as in σ-Algebra and Measurable Space (i). It is a subset of $\mathcal{P}([0,\infty])$. It contains $[0,\infty]$, since $[0,\infty]\cap\mathbb{R}=[0,\infty)\in\mathcal{B}(\mathbb{R})$. If $B\in\mathcal{D}$, then $([0,\infty]\setminus B)\cap\mathbb{R}=[0,\infty)\setminus(B\cap\mathbb{R})$, which lies in $\mathcal{B}(\mathbb{R})$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (x); so $[0,\infty]\setminus B\in\mathcal{D}$. If $(B_{n})$ is a sequence in $\mathcal{D}$, then $\bigcup_{n\in\mathbb{N}}B_{n}\subseteq[0,\infty]$ and, by Indexed Families of Sets and Their Union, Intersection and Product (ii), $\big(\bigcup_{n\in\mathbb{N}}B_{n}\big)\cap\mathbb{R}=\bigcup_{n\in\mathbb{N}}(B_{n}\cap\mathbb{R})$. Here $n\mapsto B_{n}\cap\mathbb{R}$ is a sequence in $\mathcal{P}(\mathbb{R})$ by Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (ii), whose terms lie in $\mathcal{B}(\mathbb{R})$, hence a sequence in $\mathcal{B}(\mathbb{R})$ by Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (v); so its union lies in $\mathcal{B}(\mathbb{R})$ by σ-Algebra and Measurable Space (i); so $\bigcup_{n\in\mathbb{N}}B_{n}\in\mathcal{D}$. Finally $\mathcal{R}\subseteq\mathcal{D}$: by The Borel σ-Algebra of [0,∞] (i) the elements of $\mathcal{R}$ are the rays $(a,\infty]$ with $a\in\mathbb{R}$, and for $a\in\mathbb{R}$, $(a,\infty]\cap\mathbb{R}=(a,\infty)\cap[0,\infty)$ by the preliminaries, and this set lies in $\mathcal{B}(\mathbb{R})$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (ix). By Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv), $\mathcal{B}=\sigma(\mathcal{R})\subseteq\mathcal{D}$: every $B\in\mathcal{B}$ satisfies $B\cap\mathbb{R}\in\mathcal{B}(\mathbb{R})$.

Proof of (iv). Let $\mathcal{E}$ be the set of those $C\in\mathcal{B}(\mathbb{R})$ with $C\cap[0,\infty)\in\mathcal{B}$. We check that $\mathcal{E}$ is a $\sigma$-algebra on $\mathbb{R}$. It is a subset of $\mathcal{B}(\mathbb{R})$, hence of the power set of $\mathbb{R}$. It contains $\mathbb{R}$, since $\mathbb{R}\in\mathcal{B}(\mathbb{R})$ and $\mathbb{R}\cap[0,\infty)=[0,\infty)=[0,\infty]\setminus\{\infty\}$, which lies in $\mathcal{B}$ by (ii). If $C\in\mathcal{E}$, then $\mathbb{R}\setminus C\in\mathcal{B}(\mathbb{R})$ and $(\mathbb{R}\setminus C)\cap[0,\infty)=[0,\infty)\setminus(C\cap[0,\infty))$, which lies in $\mathcal{B}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (x); so $\mathbb{R}\setminus C\in\mathcal{E}$. If $(C_{n})$ is a sequence in $\mathcal{E}$, then $\bigcup_{n\in\mathbb{N}}C_{n}\in\mathcal{B}(\mathbb{R})$, and $\big(\bigcup_{n\in\mathbb{N}}C_{n}\big)\cap[0,\infty)=\bigcup_{n\in\mathbb{N}}(C_{n}\cap[0,\infty))$ by Indexed Families of Sets and Their Union, Intersection and Product (ii). Here $n\mapsto C_{n}\cap[0,\infty]$ is a sequence in $\mathcal{P}([0,\infty])$ by Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (ii), applied with $Y=[0,\infty]$, and $C_{n}\cap[0,\infty]=C_{n}\cap[0,\infty)$ for every $n$, since $C_{n}\subseteq\mathbb{R}$ and $[0,\infty]\cap\mathbb{R}=[0,\infty)$ by the preliminaries. So $n\mapsto C_{n}\cap[0,\infty)$ is a sequence in $\mathcal{P}([0,\infty])$ whose terms lie in $\mathcal{B}$, hence a sequence in $\mathcal{B}$ by Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (v), so its union lies in $\mathcal{B}$ by σ-Algebra and Measurable Space (i); so $\bigcup_{n\in\mathbb{N}}C_{n}\in\mathcal{E}$. For $a\in\mathbb{R}$, $(a,\infty)\in\mathcal{B}(\mathbb{R})$ and $(a,\infty)\cap[0,\infty)=(a,\infty]\setminus\{\infty\}$ by the preliminaries, which lies in $\mathcal{B}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (x) and (ii); so $(a,\infty)\in\mathcal{E}$. By Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv) and Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (iii),

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

Thus $C\cap[0,\infty)\in\mathcal{B}$ for every $C\in\mathcal{B}(\mathbb{R})$. If moreover $C\subseteq[0,\infty)$, then $C=C\cap[0,\infty)\in\mathcal{B}$.

The direction from right to left of (iii). Let $B\subseteq[0,\infty]$ with $B\cap\mathbb{R}\in\mathcal{B}(\mathbb{R})$. Then $B\cap\mathbb{R}\subseteq[0,\infty]\cap\mathbb{R}=[0,\infty)$, so $B\cap\mathbb{R}\in\mathcal{B}$ by (iv). Since $[0,\infty]=[0,\infty)\cup\{\infty\}$, either $\infty\notin B$ and $B=B\cap\mathbb{R}\in\mathcal{B}$, or $\infty\in B$ and $B=(B\cap\mathbb{R})\cup\{\infty\}$, which lies in $\mathcal{B}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (vii) and (ii).

Proof of (i). Let $a\in\mathbb{R}$. By The Extended Real Line, Its Order, and [0,∞] (iii), $x>a$ means $a<x$, so $\{x\in[0,\infty]:x>a\}=(a,\infty]$ by The Borel σ-Algebra of [0,∞] (i), and this set lies in $\mathcal{B}$. The other three sets are subsets of $[0,\infty]$, and by the preliminaries and Intervals of the Real Line (iii) their intersections with $\mathbb{R}$ are

$$[a,\infty)\cap[0,\infty),\qquad (-\infty,a)\cap[0,\infty),\qquad (-\infty,a]\cap[0,\infty),$$

respectively. These lie 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) and Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (ix), so the three sets belong to $\mathcal{B}$ by (iii).