In mathematics, a Baire measure is a measure on the

σ-algebra In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...

Baire set In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalent definitions of Baire sets, but in the most ...

s 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 ...

whose value on every compact Baire set is finite. In compact

metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

the

Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...

s and the

s are the same, so Baire measures are the same as Borel measures that are finite on

compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...

s. 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

There are several inequivalent definitions of

s, 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

In practice Baire measures can be replaced by

regular Borel measure In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets. Definition Let (''X'', ''T'') be a topolo ...

s. 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

Counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ...

on the

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...

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 In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are loc ...

is a Baire measure invariant under the left (right) action of the group on itself. In particular, if the group is an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

, the left and right Haar measures coincide and we say the Haar measure is translation invariant. See also

Pontryagin duality In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...

References

* Leonard Gillman and Meyer Jerison, ''Rings of Continuous Functions'', Springer Verlag #43, 1960 {{Measure theory Measures (measure theory)