Borel set

A Borel set is any set you can build from open sets by a countable process of the basic set operations — taking complements, countable unions, countable intersections — repeated as often as you like. Intervals, open and closed sets, countable sets, and essentially every set that arises naturally in analysis are Borel. They are the “constructible” measurable sets.

Formally, the Borel sets of a topological space (in particular of R or R^n) are the members of the Borel sigma-algebra, which is the smallest sigma-algebra containing all the open sets. Every open set is Borel, hence so is every closed set (a complement), every countable intersection of open sets, every countable union of closed sets, and so on through the transfinite hierarchy. The construction is intrinsic: it does not mention any measure.

Borel sets sit strictly inside the Lebesgue measurable sets. The honest subtleties are two: not every Lebesgue measurable set is Borel — there are more Lebesgue sets than Borel sets, by a cardinality count — and although Borel sets are generated in countably many “rounds,” you genuinely need transfinitely many rounds, so a Borel set need not be a single countable union or intersection of open and closed sets.