Baire set
In mathematics, more specifically in measure theory, the notion of a Baire set is important in the understanding of particular relations between measure theory and topology. In particular, an understanding of Baire sets aids in intuition when one deals with measures on non-metrizable topological spaces. The Baire sets form a subclass of the Borel sets. The converse holds in many important, but not all, topological spaces.
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[edit] Basic definition
A subset of a compact Hausdorff topological space is called a Baire set if it is a member of the smallest σ–algebra containing all compact Gδ sets. More concisely, the Baire sets are precisely those members of the σ–algebra generated by all compact Gδ sets.
[edit] Basic example
In a Cartesian product of uncountably many compact Hausdorff spaces a Baire set is completely determined by countably many factors/co-ordinates. If each factor space in the Cartesian product has more than one point, a singleton is never a Baire set, in spite of the fact that it is closed, and therefore a Borel set.[1]
[edit] More general definitions
According to (Halmos 1950, page 220), a subset of a locally compact Hausdorff topological space is called a Baire set if it belongs to the smallest σ–ring containing all compact Gδ sets.
According to (Dudley 1989, Sect. 7.1), a subset of a topological space, X, is called a Baire set if it belongs to the smallest σ–algebra for which all continuous functions defined on X into the real line are measurable.
A discrete topological space is locally compact and Hausdorff. Any function defined on a discrete space is continuous, and therefore, according to Dudley, all subsets of a discrete space are Baire. However, since the compact subspaces of a discrete space are precisely the finite subspaces, the Baire sets, according to Halmos, are precisely the at most countable sets. Thus, the two definitions are generally non-equivalent. However, they both agree with the "basic definition" given above, if the topological space is compact Hausdorff. The rest of this article is based on Dudley's definition.
[edit] Properties
Baire sets coincide with Borel sets in every metric (or metrizable) space.[2] In particular, they coincide in Euclidean spaces and all their subsets (treated as topological spaces).
For every compact Hausdorff space, every finite Baire measure (that is, a measure on the σ-algebra of all Baire sets) is regular.[3]
For every compact Hausdorff space, every finite Baire measure has a unique extension to a regular Borel measure.[4]
The Kolmogorov extension theorem states that every consistent collection of finite-dimensional probability distributions leads to a Baire measure on the space of functions.[5] Assuming compactness one may extend it to a regular Borel measure. After completion one gets a probability space that is not necessarily standard.[6]
[edit] Notes
- ^ Dudley 1989, Example after Theorem 7.1.1
- ^ Dudley 1989, Theorem 7.1.1
- ^ Dudley 1989, Theorem 7.1.5
- ^ Dudley 1989, Theorem 7.3.1
- ^ Dudley 1989, Theorem 12.1.2
- ^ Its standardness is investigated in: Tsirelson, Boris (1981). "A natural modification of a random process and its application to stochastic functional series and Gaussian measures". Journal of Soviet Mathematics 16 (2): 940–956. doi:10.1007/BF01676139. See Theorem 1(c).
[edit] References
- Halmos, P. R. (1950). Measure theory. v. Nostrand.. See especially Sect. 51 "Borel sets and Baire sets".
- Dudley, R. M. (1989). Real Analysis and Probability. Chapman & Hall.. See especially Sect. 7.1 "Baire and Borel σ–algebras and regularity of measures" and Sect. 7.3 "The regularity extension".
