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Concept
Haar measure
Summary
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory.
Let be a locally compact Hausdorff topological group. The -algebra generated by all open subsets of is called the Borel algebra. An element of the Borel algebra is called a Borel set. If is an element of and is a subset of , then we define the left and right translates of by g as follows:
Left translate:
Right translate:
Left and right translates map Borel sets onto Borel sets.
A measure on the Borel subsets of is called left-translation-invariant if for all Borel subsets and all one has
A measure on the Borel subsets of is called right-translation-invariant if for all Borel subsets and all one has
There is, up to a positive multiplicative constant, a unique countably additive, nontrivial measure on the Borel subsets of satisfying the following properties:
The measure is left-translation-invariant: for every and all Borel sets .
The measure is finite on every compact set: for all compact .
The measure is outer regular on Borel sets :
The measure is inner regular on open sets :
Such a measure on is called a left Haar measure. It can be shown as a consequence of the above properties that for every non-empty open subset . In particular, if is compact then is finite and positive, so we can uniquely specify a left Haar measure on by adding the normalization condition .
In complete analogy, one can also prove the existence and uniqueness of a right Haar measure on . The two measures need not coincide.
Some authors define a Haar measure on Baire sets rather than Borel sets.
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