Borel sigma-algebra

The Borel sigma-algebra is the smallest possible sigma-algebra that already contains all the open sets. It is the canonical bridge between topology (which supplies the open sets) and measure theory (which needs a sigma-algebra). Whenever you want to measure subsets of a space that has a notion of openness, this is the default family to measure.

Formally, for a topological space X, the Borel sigma-algebra B(X) is the intersection of all sigma-algebras on X that contain the open sets; equivalently, it is the sigma-algebra generated by the open sets. Its members are exactly the Borel sets. The intersection of any nonempty family of sigma-algebras is again a sigma-algebra, which is why a unique smallest one exists.

On R the Borel sigma-algebra can be generated equally well by the open intervals, by the closed intervals, or by the half-lines (a, +infinity); all of these generate the same B(R). It is the natural home for Borel measures, and a real-valued function is called Borel measurable precisely when preimages of Borel sets are Borel. It is strictly smaller than the Lebesgue sigma-algebra, which adds in all subsets of Borel null sets.