Baire measure

In mathematics, a Baire measure is a measure on the σ-algebra of Baire sets of a topological space whose value on every compact Baire set is finite. In compact metric spaces the Borel sets and the Baire sets are the same, so Baire measures are the same as Borel measures that are finite on compact sets. In general Baire sets and Borel sets need not be the same. In spaces with non-Baire Borel sets, Baire measures are used because they connect to the properties of continuous functions more directly.

## Variations[edit]

There are several inequivalent definitions of Baire sets, so correspondingly there are several inequivalent concepts of Baire measure on a topological space. These all coincide on spaces that are locally compact σ-compact Hausdorff spaces.

## Relation to Borel measure[edit]

In practice Baire measures can be replaced by regular Borel measures. The relation between Baire measures and regular Borel measures is as follows:

* The restriction of a finite Borel measure to the Baire sets is a Baire measure.
* A finite Baire measure on a compact space is always regular.
* A finite Baire measure on a compact space is the restriction of a unique regular Borel measure.
* On compact (or σ-compact) metric spaces, Borel sets are the same as Baire sets and Borel measures are the same as Baire measures.

## Examples[edit]

* Counting measure on the unit interval is a measure on the Baire sets that is not regular (or σ-finite).
* The (left or right) Haar measure on a locally compact group is a Baire measure invariant under the left (right) action of the group on itself. In particular, if the group is an abelian group, the left and right Haar measures coincide and we say the Haar measure is translation invariant. See also Pontryagin duality.

## References[edit]

* Leonard Gillman and Meyer Jerison, Rings of Continuous Functions, Springer Verlag #43, 1960

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Measure theory

Basic concepts|

* Absolute continuity of measures
* Lebesgue integration
* Lp spaces
* Measure
* Measure space
* Probability space
* Measurable space/function

Sets|

* Almost everywhere
* Atom
* Baire set
* Borel set
* equivalence relation
* Borel space
* Carathéodory's criterion
* Cylindrical σ-algebra
* Cylinder set
* 𝜆-system
* Essential range
* infimum/supremum
* Locally measurable
* π-system
* σ-algebra
* Non-measurable set
* Vitali set
* Null set
* Support
* Transverse measure
* Universally measurable

Types of Measures|

* Atomic
* Baire
* Banach
* Besov
* Borel
* Brown
* Complex
* Complete
* Content
* (Logarithmically) Convex
* Decomposable
* Discrete
* Equivalent
* Finite
* Inner
* (Quasi-) Invariant
* Locally finite
* Maximising
* Metric outer
* Outer
* Perfect
* Pre-measure
* (Sub-) Probability
* Projection-valued
* Radon
* Random
* Regular
* Borel regular
* Inner regular
* Outer regular
* Saturated
* Set function
* σ-finite
* s-finite
* Signed
* Singular
* Spectral
* Strictly positive
* Tight
* Vector

Particular measures|

* Counting
* Dirac
* Euler
* Gaussian
* Haar
* Harmonic
* Hausdorff
* Intensity
* Lebesgue
* Infinite-dimensional
* Logarithmic
* Product
* Projections
* Pushforward
* Spherical measure
* Tangent
* Trivial
* Young

Maps|

* Measurable function
* Bochner
* Strongly
* Weakly
* Convergence: almost everywhere
* of measures
* in measure
* of random variables
* in distribution
* in probability
* Cylinder set measure
* Random: compact set
* element
* measure
* process
* variable
* vector
* Projection-valued measure

Main results|

* Carathéodory's extension theorem
* Convergence theorems
* Dominated
* Monotone
* Vitali
* Decomposition theorems
* Hahn
* Jordan
* Maharam's
* Egorov's
* Fatou's lemma
* Fubini's
* Fubini–Tonelli
* Hölder's inequality
* Minkowski inequality
* Radon–Nikodym
* Riesz–Markov–Kakutani representation theorem

Other results|

* Disintegration theorem
* Lifting theory
* Lebesgue's density theorem
* Lebesgue differentiation theorem
* Sard's theorem

For Lebesgue measure|

* Isoperimetric inequality
* Brunn–Minkowski theorem
* Milman's reverse
* Minkowski–Steiner formula
* Prékopa–Leindler inequality
* Vitale's random Brunn–Minkowski inequality

Applications & related|

* Convex analysis
* Descriptive set theory
* Probability theory
* Real analysis
* Spectral theory

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