Haar measure

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In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups.

This measure was introduced by Alfréd Haar, a Hungarian mathematician, in about 1932. Haar measures are used in many parts of analysis and number theory, and also in estimation theory.

[edit] Preliminaries

Let G be a locally compact topological group. In this article, the σ-algebra generated by all compact subsets of G is called the Borel algebra ^[1]. An element of the Borel algebra is called a Borel 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:

$a S = \{a \cdot s: s \in S\}.$

Right translate:

$S a = \{s \cdot a: s \in S\}.$

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 if for all Borel subsets S of G and all a in G one has

$\mu(a S) = \mu(S). \quad$

A similar definition is made for right translation invariance.

[edit] 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,^[2] we say μ is regular if and only if:

μ(K) is finite for every compact set K.

Every Borel set E is outer regular:

$\mu(E) = \inf \{\mu(U): E \subseteq U, U \mbox{ open}\}.$

Every Borel set E is inner regular:

$\mu(E) = \sup \{\mu(K): K \subseteq E, K \mbox{ compact}\}.$

The existence of Haar measure was first proven in full generality by Weil.^[3] The special case of invariant measure for compact groups had been shown by Haar in 1933.^[4]

[edit] 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 $S - 1$ the set of inverses of elements of S. If we define

$\mu_{-1}(S) = \mu(S^{-1}) \quad$

then this is a right Haar measure. To show right invariance, apply the definition:

$\mu_{-1}(S a) = \mu((S a)^{-1}) = \mu(a^{-1} S^{-1}) = \mu(S^{-1}) = \mu_{-1}(S). \quad$

Because the right measure is unique, it follows that μ_-1 is a multiple of ν and so

$\mu(S^{-1})=k\nu(S)\,$

for all Borel sets S, where k is some positive constant.

[edit] 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

$\int_G f(s x) \ d\mu(x) = \int_G f(x) \ d\mu(x)$

for any integrable function f. This is immediate for step functions, being essentially the definition of left invariance.

[edit] Uses

The Haar measures are used in harmonic analysis on arbitrary locally compact groups; see Pontryagin 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 invariant Radon 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. See non-measurable sets.

[edit] 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 (Rⁿ, +).

If G is the group of positive real numbers with multiplication as operation, then the Haar measure μ(S) is given by

$\mu(S) = \int_S \frac{1}{t} \, dt$

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

$\mu(S) = \int_S {1\over |\det(X)|^n} \, dX$

where dX denotes the Lebesgue measure on R^{$n 2$}, the set of all $n\times n$ -matrices. This follows from the change of variables formula.

More generally, on any Lie 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 the determinant of the adjoint representation.

[edit] 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

$A \mapsto \mu (t^{-1} A) \quad$

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

$\mu (t^{-1} A) = \Delta(t) \mu(A). \quad$

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

$x \mapsto a x + b\quad$

on the real line.

[edit] Notes

^ 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.
^ Paul Halmos, Measure Theory, D. van Nostrand and Co., 1950. Section 52
^ André Weil, L'intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, Hermann, 1940
^ A. Haar, Der Massbegriff in der Theorie der kontinuierlichen Gruppen, Ann. Math., v34 (1933).

[edit] Further reading

Lynn Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand and Co., 1953.
André Weil, Basic Number Theory, Academic Press, 1971.

[edit] External links

On the Existence and Uniqueness of Invariant Measures on Locally Compact Groups - by Simon Rubinstein-Salzedo

[0] 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.

[1] Paul Halmos, Measure Theory, D. van Nostrand and Co., 1950. Section 52

[2] André Weil, L'intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, Hermann, 1940

[3] A. Haar, Der Massbegriff in der Theorie der kontinuierlichen Gruppen, Ann. Math., v34 (1933).

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Haar measure

From Wikipedia, the free encyclopedia

Contents

[edit] Preliminaries

[edit] Existence and uniqueness of the left Haar measure

[edit] The right Haar measure

[edit] The Haar integral

[edit] Uses

[edit] Examples

[edit] The modular function

[edit] Notes

[edit] Further reading

[edit] External links

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