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
Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory.
Early life
He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...
in 1897 under the name "invariant integral".
Haar measures are used in many parts of
analysis,
number theory,
group theory,
representation theory,
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
,
probability theory, and
ergodic theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
.
Preliminaries
Let
be a
locally compact 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. More precisely, it is a topological space in which ev ...
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
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
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
:
Haar's theorem
There is,
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
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. This makes the regularity conditions unnecessary as Baire measures are automatically regular.
Halmos rather confusingly uses the term "Borel set" for elements of the
-ring generated by compact sets, and defines Haar measures on these sets.
The left Haar measure satisfies the inner regularity condition for all
-finite Borel sets, but may not be inner regular for ''all'' Borel sets. For example, the product of the
unit circle (with its usual topology) and the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
with the
discrete topology is a locally compact group with the
product topology and a Haar measure on this group is not inner regular for the closed subset
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
. Weil's proof used the
axiom of choice and
Henri Cartan
Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology.
He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of co ...
furnished a proof that avoided its use. Cartan's proof also establishes the existence and the uniqueness simultaneously. A simplified and complete account of Cartan's argument was given by
Alfsen in 1963. The special case of invariant measure for
second-countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
locally compact groups had been shown by Haar in 1933.
[
]
Examples
- If G is a discrete group, then the compact subsets coincide with the finite subsets, and a (left and right invariant) Haar measure on G is the counting measure.
- The Haar measure on the topological group (\mathbb, +) that takes the value 1 on the interval ,1/math> is equal to the restriction of
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
to the Borel subsets of \mathbb. This can be generalized to (\mathbb^n, +).
- In order to define a Haar measure \mu on the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
\mathbb, consider the function f from ,2\pi/math> onto \mathbb defined by f(t)=(\cos(t),\sin(t)). Then \mu can be defined by
\mu(S)=\frac1m(f^(S)),
where m is the Lebesgue measure on ,2\pi/math>. The factor (2\pi)^ is chosen so that \mu(\mathbb)=1.
- If G is the group of positive real numbers under multiplication then a Haar measure \mu is given by
\mu(S) = \int_S \frac \, dt
for any Borel subset S of positive real numbers.
For example, if S is taken to be an interval ,b/math>, then we find \mu(S) = \log(b/a). Now we let the multiplicative group act on this interval by a multiplication of all its elements by a number g, resulting in gS being the interval \cdot a,g\cdot b Measuring this new interval, we find \mu(gS) = \log((g\cdot b)/(g\cdot a)) = \log(b/a) = \mu(S).
- If G is the group of nonzero real numbers with multiplication as operation, then a Haar measure \mu is given by
\mu(S) = \int_S \frac \, dt
for any Borel subset S of the nonzero reals.
- For the general linear group G = GL(n,\mathbb), any left Haar measure is a right Haar measure and one such measure \mu is given by
\mu(S) = \int_S \, dX
where dX denotes the Lebesgue measure on \mathbb^ identified with the set of all n\times n-matrices. This follows from the change of variables formula.
- Generalizing the previous three examples, if the group G is represented as an open submanifold of \R^n with smooth group operations, then a left Haar measure on G is given by \fracd^n x, where e_1 is the group identity element of G, J_(e_1) is the Jacobian determinant of left multiplication by x at e_1, and d^n x is the Lebesgue measure on \R^n. This follows from the change of variables formula. A right Haar measure is given in the same way, except with J_(e_1) being the Jacobian of right multiplication by x.
- Let G be the set of all affine linear transformations A : \mathbb \to \mathbb of the form r \mapsto x r + y for some fixed x, y \in \mathbb with x > 0. Associate with G the operation of
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
\circ, which turns G into a non-abelian group. G can be identified with the right half plane (0, \infty) \times \mathbb = \left\ under which the group operation becomes (s, t) \circ (u, v) = (su, sv + t). A left-invariant Haar measure \mu_L (respectively, a right-invariant Haar measure \mu_R) on G = (0, \infty) \times \mathbb is given by
\mu_L(S) = \int_S \frac \,dx\,dy and \mu_R(S) = \int_S \frac \,dx\,dy
for any Borel subset S of G = (0, \infty) \times \mathbb. This is because if S \subseteq (0, \infty) \times \mathbb is an open subset then for (s, t) \in G fixed, integration by substitution gives
\mu_L((s, t) \circ S) = \int_ \frac \,dx\,dy = \int_ \frac , (s)(s) - (0)(0), \,du\,dv = \mu_L(S)
while for (u, v) \in G fixed,
\mu_R(S \circ (u, v)) = \int_ \frac \,dx\,dy = \int_S \frac , (u)(1) - (v)(0), \,ds\,dt = \mu_R(S).
- On any
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
of dimension d a left Haar measure can be associated with any non-zero left-invariant d-form \omega, as the ''Lebesgue measure'' , \omega, ; 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 unit hyperbola \ can be taken as a group under multiplication defined as with
split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s z = x + y j, \ \ j^2 = +1 . The usual area measure in the crescent C = \ serves to define hyperbolic angle as the area of its hyperbolic sector. The Haar measure of the unit hyperbola is generated by the hyperbolic angle of segments on the hyperbola. For instance, a measure of one unit is given by the segment running from (1,1) to (e,1/e), where e is Euler's number. Hyperbolic angle has been exploited in mathematical physics with rapidity
In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with di ...
standing in for classical velocity.
- If G is the group of non-zero
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s, then G can be seen as an open subset of \R^4. A Haar measure \mu is given by
\mu(S)=\int_S\frac1\,dx\,dy\,dz\,dw
where dx\wedge dy\wedge dz\wedge dw denotes the Lebesgue measure in \mathbb^4 and S is a Borel subset of G.
- If G is the additive group of p-adic numbers for a prime p, then a Haar measure is given by letting a+p^n O have measure p^, where O is the ring of p-adic integers.
Construction of Haar measure
A construction using compact subsets
The following method of constructing Haar measure is essentially the method used by Haar and Weil.
For any subsets S,T\subseteq G with S nonempty define :S/math> to be the smallest number of left translates of S that cover T (so this is a non-negative integer or infinity). This is not additive on compact sets K\subseteq G, though it does have the property that :U :U \cup L:U/math> for disjoint compact sets K,L\subseteq G provided that U is a sufficiently small open neighborhood of the identity (depending on K and L). The idea of Haar measure is to take a sort of limit of :U/math> as U becomes smaller to make it additive on all pairs of disjoint compact sets, though it first has to be normalized so that the limit is not just infinity. So fix a compact set A with non-empty interior (which exists as the group is locally compact) and for a compact set K define
:\mu_A(K)=\lim_U\frac
where the limit is taken over a suitable directed set of open neighborhoods of the identity eventually contained in any given neighborhood; the existence of a directed set such that the limit exists follows using Tychonoff's theorem.
The function \mu_A is additive on disjoint compact subsets of G, which implies that it is a regular content. From a regular content one can construct a measure by first extending \mu_A to open sets by inner regularity, then to all sets by outer regularity, and then restricting it to Borel sets. (Even for open sets U, the corresponding measure \mu_A(U) need not be given by the lim sup formula above. The problem is that the function given by the lim sup formula is not countably subadditive in general and in particular is infinite on any set without compact closure, so is not an outer measure.)
A construction using compactly supported functions
Cartan introduced another way of constructing Haar measure as a Radon measure (a positive linear functional on compactly supported continuous functions), which is similar to the construction above except that A, K, and U are positive continuous functions of compact support rather than subsets of G. In this case we define :U/math> to be the infimum of numbers c_1+\cdots+c_n such that K(g) is less than the linear combination c_1 U(g_1 g)+\cdots+c_n U(g_n g) of left translates of U for some g_1,\ldots,g_n\in G.
As before we define
:\mu_A(K)=\lim_U\frac.
The fact that the limit exists takes some effort to prove, though the advantage of doing this is that the proof avoids the use of the axiom of choice and also gives uniqueness of Haar measure as a by-product. The functional \mu_A extends to a positive linear functional on compactly supported continuous functions and so gives a Haar measure. (Note that even though the limit is linear in K, the individual terms :U/math> are not usually linear in K.)
A construction using mean values of functions
Von Neumann gave a method of constructing Haar measure using mean values of functions, though it only works for compact groups. The idea is that given a function f on a compact group, one can find a convex combination \sum a_i f(g_i g) (where \sum a_i=1) of its left translates that differs from a constant function by at most some small number \epsilon. Then one shows that as \epsilon tends to zero the values of these constant functions tend to a limit, which is called the mean value (or integral) of the function f.
For groups that are locally compact but not compact this construction does not give Haar measure as the mean value of compactly supported functions is zero. However something like this does work for almost periodic functions on the group which do have a mean value, though this is not given with respect to Haar measure.
A construction on Lie groups
On an ''n''-dimensional Lie group, Haar measure can be constructed easily as the measure induced by a left-invariant ''n''-form. This was known before Haar's theorem.
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 \nu satisfying the above regularity conditions and being finite on compact sets, but it need not coincide with the left-translation-invariant measure \mu. The left and right Haar measures are the same only for so-called ''unimodular groups'' (see below). It is quite simple, though, to find a relationship between \mu and \nu.
Indeed, for a Borel set S, let us denote by S^ the set of inverses of elements of S. If we define
: \mu_(S) = \mu(S^) \quad
then this is a right Haar measure. To show right invariance, apply the definition:
: \mu_(S g) = \mu((S g)^) = \mu(g^ S^) = \mu(S^) = \mu_(S). \quad
Because the right measure is unique, it follows that \mu_ is a multiple of \nu and so
:\mu(S^)=k\nu(S)\,
for all Borel sets S, where k is some positive constant.
The modular function
The ''left'' translate of a right Haar measure is a right Haar measure. More precisely, if \nu is a right Haar measure, then for any fixed choice of a group element ''g'',
: S \mapsto \nu (g^ S) \quad
is also right invariant. Thus, by uniqueness up to a constant scaling factor of the Haar measure, there exists a function \Delta from the group to the positive reals, called the Haar modulus, modular function or modular character, such that for every Borel set S
: \nu (g^ S) = \Delta(g) \nu(S). \quad
Since right Haar measure is well-defined up to a positive scaling factor, this equation shows the modular function is independent of the choice of right Haar measure in the above equation.
The modular function is a continuous group homomorphism from ''G'' to the multiplicative group of positive real numbers. A group is called unimodular if the modular function is identically 1, or, equivalently, if the Haar measure is both left and right invariant. Examples of unimodular groups are abelian groups, compact groups, discrete groups (e.g., finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or ma ...
s), semisimple Lie groups and connected nilpotent Lie groups. An example of a non-unimodular group is the group of affine transformations
:\big\=\left\
on the real line. This example shows that a solvable Lie group need not be unimodular.
In this group a left Haar measure is given by \fracda\wedge db, and a right Haar measure by \fracda\wedge db.
Measures on homogeneous spaces
If the locally compact group G acts transitively on a homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
G/H, one can ask if this space has an invariant measure, or more generally a semi-invariant measure with the property that \mu(gS) = \chi(g)\mu(S) for some character \chi of G. A necessary and sufficient condition for the existence of such a measure is that the restriction \chi, _H is equal to \Delta, _H/\delta, where \Delta and \delta are the modular functions of G and H respectively.
In particular an invariant measure on G/H exists if and only if the modular function \Delta of G restricted to H is the modular function \delta of H.
Example
If G is the group SL_2(\mathbb) and H is the subgroup of upper triangular matrices, then the modular function of H is nontrivial but the modular function of G is trivial. The quotient of these cannot be extended to any character of G, so the quotient space G/H (which can be thought of as 1-dimensional real projective space) does not have even a semi-invariant measure.
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 and is denoted as:
:\int f(x) \, d\mu(x)
where \mu is the Haar measure.
One property of a left Haar measure \mu is that, letting s be an element of G, the following is valid:
: \int_G f(sx) \ d\mu(x) = \int_G f(x) \ d\mu(x)
for any Haar integrable function f on G. This is immediate for indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
s:
: \int \mathit_A(tg)\,d\mu = \int \mathit_(g)\,d\mu=\mu(t^A)=\mu(A)=\int\mathit_A(g)\,d\mu,
which is essentially the definition of left invariance.
Uses
In the same issue of '' Annals of Mathematics'' and immediately after Haar's paper, the Haar theorem was used to solve Hilbert's fifth problem restricted to compact groups by John von Neumann.
Unless G is a discrete group, it is impossible to define a countably additive left-invariant regular measure on ''all'' subsets of G, assuming the axiom of choice, according to the theory of non-measurable sets.
Abstract harmonic analysis
The Haar measures are used in harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
on locally compact groups, particularly in the theory of Pontryagin duality. To prove the existence of a Haar measure on a locally compact group G it suffices to exhibit a left-invariant Radon measure on G.
Mathematical statistics
In mathematical statistics, Haar measures are used for prior measures, which are prior probabilities for compact groups of transformations. These prior measures are used to construct admissible procedures, by appeal to the characterization of admissible procedures as Bayesian procedures (or limits of Bayesian procedures) by Wald. For example, a right Haar measure for a family of distributions with a location parameter results in the Pitman estimator
In statistics, the concept of being an invariant estimator is a criterion that can be used to compare the properties of different estimators for the same quantity. It is a way of formalising the idea that an estimator should have certain intuitive ...
, which is best
Best or The Best may refer to:
People
* Best (surname), people with the surname Best
* Best (footballer, born 1968), retired Portuguese footballer
Companies and organizations
* Best & Co., an 1879–1971 clothing chain
* Best Lock Corporation ...
equivariant. When left and right Haar measures differ, the right measure is usually preferred as a prior distribution. For the group of affine transformations on the parameter space of the normal distribution, the right Haar measure is the Jeffreys prior measure. Unfortunately, even right Haar measures sometimes result in useless priors, which cannot be recommended for practical use, like other methods of constructing prior measures that avoid subjective information.
Another use of Haar measure in statistics is in conditional inference The conditionality principle is a Fisherian principle of statistical inference that Allan Birnbaum formally defined and studied in his 1962 JASA article. Informally, the conditionality principle can be taken as the claim that experiments which were ...
, in which the sampling distribution of a statistic is conditioned on another statistic of the data. In invariant-theoretic conditional inference, the sampling distribution is conditioned on an invariant of the group of transformations (with respect to which the Haar measure is defined). The result of conditioning sometimes depends on the order in which invariants are used and on the choice of a maximal invariant
Maximal may refer to:
*Maximal element, a mathematical definition
*Maximal (Transformers), a faction of Transformers
*Maximalism, an artistic style
*Maximal set
* ''Maxim'' (magazine), a men's magazine marketed as ''Maximal'' in several countries
...
, so that by itself a statistical principle of invariance fails to select any unique best conditional statistic (if any exist); at least another principle is needed.
For non-compact groups, statisticians have extended Haar-measure results using amenable groups.
Weil's converse theorem
In 1936, André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
proved a converse (of sorts) to Haar's theorem, by showing that if a group has a left invariant measure with a certain ''separating'' property,[ then one can define a topology on the group, and the completion of the group is locally compact and the given measure is essentially the same as the Haar measure on this completion.
]
See also
* Invariant measure
* Pontryagin duality
* Riesz–Markov–Kakutani representation theorem
In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for who introduced it for continuou ...
Notes
Further reading
*
*.
*
*
* André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
, ''Basic Number Theory'', Academic Press, 1971.
External links
The existence and uniqueness of the Haar integral on a locally compact topological group
- by Gert K. Pedersen
On the Existence and Uniqueness of Invariant Measures on Locally Compact Groups
- by Simon Rubinstein-Salzedo
{{DEFAULTSORT:Haar Measure
Lie groups
Topological groups
Measures (measure theory)
Harmonic analysis