HOME

TheInfoList



OR:

In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that avoids some of the pathological properties of Borel sets. There are several inequivalent definitions of Baire sets, but in the most widely used, the Baire sets of a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
form the smallest σ-algebra such that all compactly supported continuous functions are measurable. Thus, measures defined on this σ-algebra, called Baire 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 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 The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...
Borel measures on Borel sets. Baire sets were introduced by , and , who named them after Baire functions, which are in turn named after René-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 compact σ-compact Hausdorff 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 Kodaira defined 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). 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.


Second definition

defined Baire sets of a locally compact Hausdorff space to be the elements of the σ-ring generated by the compact ''G''δ sets. 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 compact ''G''δ sets rather than closed ''G''δ sets is that Baire measures are then automatically
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...
.


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 a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
Hausdorff topological space is called a ''Baire set'' if it is a member of the smallest σ–algebra containing all compact ''G''δ sets. In other words, the σ–algebra of Baire sets is the σ–algebra ''generated'' by all compact ''G''δ sets. 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 Baire
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...
that 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. A
discrete topological space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest t ...
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''δ compact sets. For example, if ''X'' is an infinite countable set whose closed sets are the finite sets and the whole space, then the only continuous real functions on ''X'' are constant, but all subsets of ''X'' are in the σ-algebra generated by compact closed ''G''δ sets.


A Borel set that is not a Baire set

In a Cartesian product of uncountably many compact
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
s 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.


Properties

Baire sets coincide with Borel sets in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
s. For every compact Hausdorff space, every finite Baire measure (that is, a measure on the σ-algebra of all Baire sets) is
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...
. For every compact Hausdorff space, every finite Baire measure has a unique extension to a regular Borel measure. The Kolmogorov extension theorem states that every consistent collection of finite-dimensional probability distributions leads to a Baire measure on the space of functions. 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.Its standardness is investigated in: . See Theorem 1(c).


Notes


References

* See especially Sect. 51 "Borel sets and Baire sets". * . See especially Sect. 7.1 "Baire and Borel σ–algebras and regularity of measures" and Sect. 7.3 "The regularity extension". * * * {{Measure theory General topology Measure theory