 Solution

The Borel sigma-algebra $\mathcal{B}(\mathbb{R})$ is the smallest sigma-algebra containing all open subsets of $\mathbb{R}$. A real-valued Borel measurable function on $(E,\mathcal{E})$ is a map $f$ with $f^{-1}(B)\in \mathcal{E}$ for every $B\in \mathcal{B}(\mathbb{R})$. It suffices to test the sets $(a,\infty )$ for rational $a$, because these generate the Borel sigma-algebra.

Here “Borel function” describes measurability into the Borel structure of the target; the domain need not itself be a topological space.