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.

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.

1 Preliminaries
2 Existence and uniqueness of the left Haar measure
3 The right Haar measure
4 The Haar integral
5 Uses
6 Examples
7 The modular function
8 Notes and references
9 Additional references
10 External links

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.

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].

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.

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.

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.

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, 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.

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.

Notes and references

↑ 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).

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

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

Integrals

Methods	Riemann integral · Lebesgue integral · Bochner integral · Daniell integral · Darboux integral · Henstock-Kurzweil integral · Haar integral · Lebesgue-Stieltjes integral · Pfeffer integral · Riemann-Stieltjes integral

Improper Integrals	Improper integral · Gaussian integral

Stochastic integrals	Itō integral · Stratonovich integral