Baire set
In mathematics, more specifically in measure theory, the Baire sets form a
There are several inequivalent definitions of Baire sets, but in the most widely used, the Baire sets of a locally compact Hausdorff space form the smallest
Every Baire set is a Borel set. The converse holds in many, but not all, topological spaces. Baire sets avoid some pathological properties of Borel sets on spaces without a countable base for the topology. In practice, the use of Baire measures on Baire sets can often be replaced by the use of regular Borel measures on Borel sets.
Baire sets were introduced by Kunihiko Kodaira (1941, Definition 4), Shizuo Kakutani and Kunihiko Kodaira (1944) and Halmos (1950, page 220), who named them after Baire functions, which are in turn named after René-Louis Baire.
Basic definitions
[edit]There are at least three inequivalent definitions of Baire sets on locally compact Hausdorff spaces, and even more definitions for general topological spaces, though all these definitions are equivalent for locally compact
First definition
[edit]Kunihiko Kodaira defined[1] what we call Baire sets (although he confusingly calls them "Borel sets") of certain topological spaces to be the sets whose characteristic function is a Baire function (the smallest class of functions containing all continuous real-valued functions and closed under pointwise limits of sequences).
Dudley (1989, Sect. 7.1) gives an equivalent definition and defines Baire sets of a topological space to be elements of the smallest
Conversely, the Baire functions are exactly the real-valued functions that are Baire measurable. For metric spaces, the Baire sets coincide with the Borel sets.[2]
Second definition
[edit]Halmos (1950, page 220) defined Baire sets of a locally compact Hausdorff space to be the elements of the
One reason for working with compact G
Third definition
[edit]The third and most widely used definition is similar to Halmos's definition, modified so that the Baire sets form a
A subset of a locally compact Hausdorff topological space is called a Baire set if it is a member of the smallest
For
Examples
[edit]The different definitions of Baire sets are not equivalent
[edit]For locally compact Hausdorff topological spaces that are not
A discrete topological space is locally compact and Hausdorff. Any function defined on a discrete space is continuous, and therefore, according to the first definition, 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 the second definition, are precisely the at most countable sets, while according to the third definition the Baire sets are the at most countable sets and their complements. Thus, the three definitions are non-equivalent on an uncountable discrete space.
For non-Hausdorff spaces the definitions of Baire sets in terms of continuous functions need not be equivalent to definitions involving G
A Borel set that is not a Baire set
[edit]In a Cartesian product of uncountably many compact Hausdorff spaces with more than one point, a point is never a Baire set, in spite of the fact that it is closed, and therefore a Borel set.[3]
Properties
[edit]Baire sets coincide with Borel sets in Euclidean spaces.
For every compact Hausdorff space, every finite Baire measure (that is, a measure on the
For every compact Hausdorff space, every finite Baire measure has a unique extension to a regular Borel measure.[5]
The Kolmogorov extension theorem states that every consistent collection of finite-dimensional probability distributions leads to a Baire measure on the space of functions.[6] Assuming compactness (of the given space, and therefore also the function space) one may extend it to a regular Borel measure. After completion one gets a probability space that is not necessarily standard.[7]
Notes
[edit]- ^ Kodaira 1941, p. 21, Def. 4
- ^ Dudley (1989, Theorem 7.1.1)
- ^ Dudley 1989, Example after 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. S2CID 119874519.. See Theorem 1(c).
References
[edit]- 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. ISBN 0521007542.. See especially Sect. 7.1 "Baire and Borel
σ –algebras and regularity of measures" and Sect. 7.3 "The regularity extension". - Kakutani, Shizuo; Kodaira, Kunihiko (1944), "Über das Haarsche Mass in der lokal bikompakten Gruppe", Proc. Imp. Acad. Tokyo, 20 (7): 444–450, doi:10.3792/pia/1195572875, MR 0014401
- Kodaira, Kunihiko (1941), "Über die Gruppe der messbaren Abbildungen", Proc. Imp. Acad. Tokyo, 17: 18–23, doi:10.3792/pia/1195578914, MR 0004089
- "Baire set", Encyclopedia of Mathematics, EMS Press, 2001 [1994]