Baire subset



Baire sets are certain subsets of a topological space. They form the Baire σ\sigma-algebra of the space, and they play an important role in measure theory.


Let XX be a topological space. Then there is a σ\sigma-algebra \mathcal{B} on XX generated by the open subsets of XX that are preimages of (0,)(0,\infty) under some continuous map XRX\to\mathbf{R}. Elements of \mathcal{B} are called the Baire sets (or Baire subsets, or Baire-measurable sets, etc) of XX, and \mathcal{B} itself is called the Baire σ\sigma-algebra on XX.


The Baire σ\sigma-algebra is a σ\sigma-subalgebra of the Borel σ-algebra since every continuous map is measurable. Often both σ-algebras even coincide. This holds for perfectly normal spaces, such as metric spaces or regular hereditary Lindelöf spaces.

When working with locally compact spaces one can often instead use the following fact (Dudley, Theorem 7.3.1):


Let K K be a compact Hausdorff space and μ \mu any finite Baire measure thereon. Then μ \mu has a unique extension to a regular Borel measure on K K .