Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line

Let $\mathbb{Q}$ be the set of rational numbers and let $\mathcal{B}(\mathbb{R})$ be the Borel $\sigma$-algebra on the real line. Write $\mathcal{I}$ for the family of open intervals $(p,q)$ with $p\in\mathbb{Q}$ and $q\in\mathbb{Q}$, and $\mathcal{R}$ for the family of rays $\{x\in\mathbb{R}:a<x\}$ with $a\in\mathbb{R}$.

1. (Rational exhaustion of open sets) For every Euclidean open subset $U$ of $\mathbb{R}$ there is a sequence $(I_k)_{k\in\mathbb{N}}$ of members of $\mathcal{I}$ with

2. (Generators) $\mathcal{B}(\mathbb{R})$ is the $\sigma$-algebra generated by $\mathcal{I}$, and it is also the $\sigma$-algebra generated by $\mathcal{R}$.

3. (Criterion for real-valued measurability) Let $(X,\mathcal{F})$ be a measurable space and let $f:X\to\mathbb{R}$. Then $f$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R})$ if and only if

for every real number $a$.