Baire set
Encyclopedia
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 space
s. The Baire sets form a subclass of the Borel sets. The converse holds in many important, but not all, topological spaces.
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.
Hausdorff space
s 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.
Hausdorff topological space is called a Baire set if it belongs to the smallest σ–ring containing all compact Gδ sets
.
According to , a subset of a topological space, X, is called a Baire set if it belongs to the smallest σ–algebra
for which all continuous function
s 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.
(or metrizable) space. In particular, they coincide in Euclidean space
s 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
.
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 one may extend it to a regular Borel measure. After completion
one gets a probability space that is not necessarily standard
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, more specifically in measure theory, the notion of a Baire set is important in the understanding of particular relations between measure theory
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
and topology
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
. In particular, an understanding of Baire sets aids in intuition when one deals with measures on non-metrizable topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s. The Baire sets form a subclass of the Borel sets. The converse holds in many important, but not all, topological spaces.
Basic definition
A subset of a compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
topological space is called a Baire set if it is a member of the smallest σ–algebra
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
containing all compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Gδ sets
G-delta set
In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with G for Gebiet meaning open set in this case and δ for Durchschnitt .The term inner limiting set is also used...
. More concisely, the Baire sets are precisely those members of the σ–algebra generated by all compact Gδ sets.
Basic example
In a Cartesian product of uncountably many compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
s 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.
More general definitions
According to , a subset of a locally compactLocally compact space
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.-Formal definition:...
Hausdorff topological space is called a Baire set if it belongs to the smallest σ–ring containing all compact Gδ sets
G-delta set
In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with G for Gebiet meaning open set in this case and δ for Durchschnitt .The term inner limiting set is also used...
.
According to , a subset of a topological space, X, is called a Baire set if it belongs to the smallest σ–algebra
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
for which all continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s 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
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
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.
Properties
Baire sets coincide with Borel sets in every metricMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
(or metrizable) space. In particular, they coincide in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
s 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
Regular measure
In mathematics, a regular measure on a topological space is a measure for which every measurable set is "approximately open" and "approximately closed".-Definition:...
.
For every compact Hausdorff space, every finite Baire measure has a unique extension to a regular Borel
measure.
The Kolmogorov extension theorem
Kolmogorov extension theorem
In mathematics, the Kolmogorov extension theorem is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process...
states that every consistent collection of finite-dimensional probability distributions leads to a Baire measure on the space of functions. Assuming compactness one may extend it to a regular Borel measure. After completion
Complete measure
In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...
one gets a probability space that is not necessarily standard
Standard probability space
In probability theory, a standard probability space is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940...
.