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Encyclopedia > Haar measure

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. Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...

This measure was introduced by Alfréd Haar, a Hungarian mathematician, about 1932. Haar measures are used in many parts of analysis and number theory. In mathematics, a measure is a function that assigns a number, e. ... AlfrÃ©d Haar (October 11, 1885 - March 16, 1933) was a Hungarian mathematician. ... This article is in need of attention from an expert on the subject. ... 1932 (MCMXXXII) is a leap year starting on Friday. ...

1 Preliminaries
2 Existence of the left Haar measure
3 The right Haar measure
4 The Haar integral
5 Uses
6 Examples
7 The modular function
8 References
9 See also

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. 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: In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... In mathematics, the Borel algebra (or Borel Ïƒ-algebra) on a topological space X is a Ïƒ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this Ïƒ-algebra: The minimal Ïƒ-algebra containing the open sets. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ...

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.

Existence 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. Here, following Halmos, Section 52, we say μ is regular iff: Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... â†” â‡” â‰¡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

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

Every Borel set E is outer regular:

If E is Borel, then E is inner regular:

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

Remark. In some pathological cases, a set can be open without being Borel. For this reason, in the property of outer regularity, the range of the infimum is specifically stated to be over sets which are open and Borel. These pathologies never occur if G is a locally compact group whose underlying topology is separable metric; in this case the Borel structure is that generated by all open sets.

The right Haar measure

It can also be proved that there exists an essentially unique 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

$mu(S^{-1})=knu(S),$

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 The integral can be interpreted as the area under a curve. ...

$int_G f(s x) dmu(x) = int_G f(x) dmu(x)$

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, 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. Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ...

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. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In mathematics, a non-measurable set is a set whose structure is so complicated it sheds light on the very notion of length, area or volume. ...

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

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

where dX denotes the Lebesgue measure on R^{$n 2$}, the set of all $ntimes n$ -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.

In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... The adjoint representation of a Lie group G is the linearized version of the action of G on itself by conjugation. ...

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 modular function such that for every Borel set A

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

A group is unimodular iff 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

$x mapsto a x + bquad$

on the real line.

References

Paul Halmos, Measure Theory, D. van Nostrand and Co., 1950.
Lynn Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand and Co., 1953.
André Weil, Basic Number Theory, Academic Press, 1971

Paul Halmos Paul Richard Halmos (born March 3, 1916) is a Hungarian-born American mathematician who has done research in the fields of logarithm theory, probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). ... AndrÃ© Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ...

Contents

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