nLab Baire subset

Idea

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

Definition

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

Properties

The Baire $\sigma$-algebra is a $\sigma$-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):

Theorem

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

Related concepts

• Borel set

• Baire measure?

References

• Dudley, Real analysis and probability, 2002.