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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The course is based on Durrett's text book
Probability: Theory and Examples.
It takes the measure theory approach to probability theory, wherein expectations are simply abstract integrals.
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups.
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.
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 locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on G so that standard analysis notions such as the Fourier transform and spaces can be generalized.
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