Borel Sigma-Algebra

The Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ on $\mathbb{R}$ is the smallest $\sigma$-algebra (Definition~Sigma-Algebra) containing all open subsets of $\mathbb{R}$. More generally, for a topological space $X$, the Borel $\sigma$-algebra $\mathcal{B}(X)$ is generated by the open sets of $X$.