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Wikipedia / Haar measure

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 in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory.

Preliminaries

be a locally compact Hausdorff topological group. The \sigma-algebra is called the Borel algebra. An element of the Borel algebra is called a Borel set, then we define the left and right translates as follows:

Left and right translates map Borel sets onto Borel sets.

is called left-translation-invariant is called right-translation-invariant

Haar's theorem

There is, up to a positive multiplicative constant, a unique countably additive satisfying the following properties:

is called a left Haar measure..

In complete analogy, one can also prove the existence and uniqueness of a right Haar measure. 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 uses the nonstandard term "Borel set" for elements of the \sigma-ring generated by compact sets, and defines Haar measures on these sets.

The left Haar measure satisfies the inner regularity condition for all \sigma-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 with the discrete topology is a locally compact group with the product topology. But, using outer regularity, one can show the segment has infinite measure.)

The existence and uniqueness (up to scaling) of a left Haar measure was first proven in full generality by André Weil. Weil's proof used the axiom of choice and Henri Cartan 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 locally compact groups had been shown by Haar in 1933.

Examples

is a discrete group is the counting measure. is equal to the restriction of Lebesgue measure on the circle group. is the group of positive real numbers of the nonzero reals. For the general linear group-matrices. This follows from the change of variables formula. with smooth is the Jacobian determinant. This follows from the change of variables formula. For the orthogonal group, that is, a matrix with all entries being IID samples of the normal distribution with mean zero and variance one. Next use Gram–Schmidt process and it is distributed according to the probability Haar measure on that group. Since the special orthogonal group (in random variable terms this means conditioning the determinant to be 1, an event of probability 1/2). can be used to construct the Haar measure on the unitary group. For the special unitary group. the operation of function composition and fixed, integration by substitution On any Lie group a left Haar measure can be associated with any non-zero left-invariant d-form, as the Lebesgue measure; 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. A representation of the Haar measure of positive real numbers in terms of area under the positive branch of the standard hyperbola xy = 1 uses Borel sets generated by intervals [a,b], b > a > 0. For example, a = 1 and b = Euler’s number e yields and area equal to log (e/1) = 1. Then for any positive real number c the area over the interval [ca, cb] equals log (b/a) so the area is invariant under multiplication by positive real numbers. Note that the area approaches infinity both as a approaches zero and b gets large. Use of this Haar measure to define a logarithm function anchors a at 1 and considers area over an interval in [b,1], with 0 < b < 1, as negative area. In this way the logarithm can take any real value even though measure is always positive or zero. is the group of non-zero quaternions. is the additive group of p-adic numbers-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.

define

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.

, which implies that it is a regular content 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. As before we define

  1. .

.)

A construction using mean values of functions

on a compact group, one can find a convex combination.

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

. The left and right Haar measures are the same only for so-called unimodular groups. If we define

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

and so

is some positive constant.

The modular function

The left is a right Haar measure, then for any fixed choice of a group element g,

from the group to the positive reals, called the Haar modulus, modular function or modular character

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, 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 groups), semisimple Lie groups and connected nilpotent Lie groups. An example of a non-unimodular group is the group of affine transformations

on the real line. This example shows that a solvable.

Measures on homogeneous spaces

acts transitively on a homogeneous space respectively..

Example

(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. This integral is called the Haar integral and is denoted as:

is the Haar measure.

, the following is valid:

. This is immediate for indicator functions:

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.

is a discrete group, it is impossible to define a countably additive left-invariant regular measure on all, assuming the axiom of choice, according to the theory of non-measurable sets.

Abstract harmonic analysis

The Haar measures are used in harmonic analysis on locally compact groups, particularly in the theory of Pontryagin duality. it suffices to exhibit a left-invariant Radon measure.

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, which is best 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, 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, 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 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

Notes

References