- Haar measure
In
mathematical analysis , the Haar measure is a way to assign an "invariant volume" to subsets oflocally compact topological group s and subsequently define anintegral for functions on those groups.This measure was introduced by
Alfréd Haar , a Hungarianmathematician , in about1932 . Haar measures are used in many parts of analysis andnumber theory , and also inestimation theory .Preliminaries
Let "G" be a locally compact
topological group . In this article, the σ-algebra generated by all compact subsets of "G" is called theBorel algebra [ We follow the conventions of Halmos' textbook. Many authors instead use the term Borel algebra to denote the σ-algebra generated by the open sets.] . An element of the Borel algebra is called aBorel set . If "a" is an element of "G" and "S" is a subset of "G", then we define the left and right translates of "S" as follows:
* Left translate::
* Right translate::Left and right translates map Borel sets into Borel sets.
A measure μ on the Borel subsets of "G" is called "left-translation-invariant" if and only iffor all Borel subsets "S" of "G" and all "a" in "G" one has:A similar definition is made for right translation invariance.
Existence and uniqueness of the left Haar measure
It turns out that there is,
up to a positive multiplicative constant, only one left-translation-invariant countably additive regular measure μ on the Borel subsets of "G" such that μ("U") > 0 for any open non-empty Borel set "U". Such a measure is called a "left Haar measure." Following Halmos [ Paul Halmos, "Measure Theory", D. van Nostrand and Co., 1950.Section 52] , we say μ is regularif and only if :* μ("K") is finite for every compact set "K".
* Every Borel set "E" is outer regular:::
* Every Borel set "E" is inner regular:::
The existence of Haar measure was first proven in full generality by Weil [André Weil, "L'intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, Hermann, 1940"] . The special case of invariant measure for compact groups had been shown by Haar in 1933 [ A. Haar, "Der Massbegriff in der Theorie der kontinuierlichen Gruppen", Ann. Math., v34 (1933).] .
The right Haar measure
It can also be proved that there exists a unique (up to multiplication by a positive constant) right-translation-invariant Borel measure ν, but it need not coincide with the left-translation-invariant measure μ. These measures are the same only for so-called "unimodular groups" (see below). It is quite simple though to find a relationship between μ and ν.
Indeed, for a Borel set "S", let us denote by the set of inverses of elements of "S". If we define :then this is a right Haar measure. To show right invariance, apply the definition:
:
Because the right measure is unique, it follows that μ-1 is a multiple of ν and so:for all Borel sets "S", where "k" is some positive constant.
The Haar integral
Using the general theory of
Lebesgue integration , one can then define an integral for all Borel measurable functions "f" on "G". This integral is called the Haar integral. If μ is a left Haar measure, then:for any integrable function "f". This is immediate for step functions, being essentially the definition of left invariance.Uses
The Haar measures are used in
harmonic analysis on arbitrary locally compact groups, seePontryagin duality . A frequently used technique for proving the existence of a Haar measure on a locally compact group "G" is showing the existence of a left invariantRadon measure on "G".Unless "G" is a discrete group, it is impossible to define a countably-additive right invariant measure on "all" subsets of "G", assuming the
axiom of choice . Seenon-measurable set s.Examples
* The Haar measure on the topological group (R, +) which takes the value 1 on the interval [0,1] is equal to the restriction of
Lebesgue measure to the Borel subsets of "R". This can be generalized for (R"n", +).* If "G" is the group of positive real numbers with multiplication as operation, then the Haar measure μ("S") is given by:::for any Borel subset "S" of the positive reals.
This generalizes to the following:
* For "G" = "GL"(n,R), left and right Haar measures are proportional and:::where "dX" denotes the Lebesgue measure on R, the set of all -matrices. This follows from thechange of variables formula .
* More generally, on anyLie group of dimension "d" a left Haar measure can be associated with any non-zero left-invariant "d"-form ω, as the "Lebesgue measure" |ω|; and similarly for right Haar measures. This means also that the modular function can be computed, as the absolute value of thedeterminant of theadjoint representation .The modular function
The "left" translate of a right Haar measure is a right Haar measure. More precisely, if μ is a right Haar measure, then
:
is also right invariant. Thus, there exists a unique function Δ called the Haar modulus, modular function or modular character, such that for every Borel set "A"
:
Note that the modular function is a group homomorphism into the multiplicative group of nonzero real numbers. A group is unimodular if and only if the modular function is identically 1. Examples of unimodular groups are compact groups and abelian groups. An example of a non-unimodular group is the "ax" + "b" group of transformations of the form
:
on the real line.
Notes and references
Additional references
* Lynn Loomis, "An Introduction to Abstract Harmonic Analysis", D. van Nostrand and Co., 1953.
*André Weil , "Basic Number Theory", Academic Press, 1971.External links
* [http://www.artofproblemsolving.com/LaTeX/Examples/HaarMeasure.pdf On the Existence and Uniqueness of Invariant Measures on Locally Compact Groups] - by Simon Rubinstein-Salzedo
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