Baire set

Inmathematics,more specifically inmeasure theory,theBaire setsform aσ-algebraof atopological spacethat avoids some of the pathological properties ofBorel sets.

There are several inequivalent definitions of Baire sets, but in the most widely used, the Baire sets of alocally compact Hausdorff spaceform the smallest σ-algebra such that allcompactly supportedcontinuous functions aremeasurable.Thus, measures defined on this σ-algebra, calledBaire measures,are a convenient framework for integration on locally compact Hausdorff spaces. In particular, any compactly supported continuous function on such a space is integrable with respect to any finite Baire measure.

Every Baire set is aBorel 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 ofregularBorel 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 afterBaire functions,which are in turn named afterRené-Louis Baire.

Basic definitions

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 compactσ-compactHausdorff spaces. Moreover, some authors add restrictions on the topological space that Baire sets are defined on, and only define Baire sets on spaces that are compact Hausdorff, or locally compact Hausdorff, or σ-compact.

First definition

Kunihiko Kodairadefined^[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 σ-algebra such that all continuous real-valued functions are measurable. For locally compact σ-compact Hausdorff spaces this is equivalent to the following definitions, but in general the definitions are not equivalent.

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

Halmos (1950,page 220) defined Baire sets of a locally compact Hausdorff space to be the elements of theσ-ringgenerated by the compactG_δsets. This definition is no longer used much, as σ-rings are somewhat out of fashion. When the space is σ-compact, this definition is equivalent to the next definition.

One reason for working with compactG_δsets rather than closedG_δsets is that Baire measures are then automaticallyregular(Halmos 1950,theorem G page 228).

Third definition

The third and most widely used definition is similar to Halmos's definition, modified so that the Baire sets form a σ-algebra rather than just a σ-ring.

A subset of alocally compact Hausdorfftopological space is called aBaire setif it is a member of the smallestσ–algebracontaining allcompact G_δsets.In other words, the σ–algebra of Baire sets is the σ–algebrageneratedby all those intersections of countably many open sets that yield a compact set. Alternatively, Baire sets form the smallest σ-algebra such that all continuous functions of compact support are measurable (at least on locally compact Hausdorff spaces; on general topological spaces these two conditions need not be equivalent).

For σ-compact spaces this is equivalent to Halmos's definition. For spaces that are not σ-compact the Baire sets under this definition are those under Halmos's definition together with their complements. However, in this case it is no longer true that a finite Baire measure is necessarily regular: for example, the Baireprobability measurethat assigns measure 0 to every countable subset of an uncountable discrete space and measure 1 to every co-countable subset is a Baire probability measure that is not regular.

Examples

The different definitions of Baire sets are not equivalent

For locally compact Hausdorff topological spaces that are not σ-compact the three definitions above need not be equivalent.

Adiscrete topological spaceis 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 theat most countablesets, 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 involvingG_δcompact sets. For example, ifXis an infinite countable set whose closed sets are the finite sets and the whole space, then the only continuous real functions onXare constant, but all subsets ofXare in the σ-algebra generated by compact closedG_δsets.

A Borel set that is not a Baire set

In a Cartesian product of uncountably manycompact Hausdorff spaceswith 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

Baire sets coincide with Borel sets inEuclidean spaces.

For every compact Hausdorff space, every finite Baire measure (that is, a measure on the σ-algebra of all Baire sets) isregular.^[4]

For every compact Hausdorff space, every finite Baire measure has a unique extension to a regular Borel measure.^[5]

TheKolmogorov extension theoremstates 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, andtherefore also the function space) one may extend it to a regular Borel measure. Aftercompletionone gets a probability space that is not necessarilystandard.^[7]

Notes

^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

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]

[1] Kodaira 1941,p. 21, Def. 4

[2] Dudley (1989,Theorem 7.1.1)

[3] Dudley 1989,Example after Theorem 7.1.1

[4] Dudley 1989,Theorem 7.1.5

[5] Dudley 1989,Theorem 7.3.1

[6] Dudley 1989,Theorem 12.1.2

[7] 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).

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