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

Let be a locally compact 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 of by 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 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 is nontrivial if and only if 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 uses the nonstandard 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 and the real line 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.
The existence and uniqueness 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

If is a discrete group, then the compact subsets coincide with the finite subsets, and a Haar measure on is the counting measure.
The Haar measure on the topological group that takes the value on the interval is equal to the restriction of Lebesgue measure to the Borel subsets of. This can be generalized to
In order to define a Haar measure on the circle group, consider the function from onto defined by. Then can be defined by
where is the Lebesgue measure on. The factor is chosen so that.
If is the group of positive real numbers under multiplication then a Haar measure is given by
for any Borel subset of positive real numbers.
For example, if is taken to be an interval, then we find. Now we let the multiplicative group act on this interval by a multiplication of all its elements by a number, resulting in being the interval Measuring this new interval, we find
If is the group of nonzero real numbers with multiplication as operation, then a Haar measure is given by
for any Borel subset of the nonzero reals.
For the general linear group, any left Haar measure is a right Haar measure and one such measure is given by
where denotes the Lebesgue measure on identified with the set of all -matrices. This follows from the change of variables formula.
Generalizing the previous three examples, if the group is represented as an open submanifold of with smooth group operations, then a left Haar measure on is given by, where is the group identity element of, is the Jacobian determinant of left multiplication by at, and is the Lebesgue measure on. This follows from the change of variables formula. A right Haar measure is given in the same way, except with being the Jacobian of right multiplication by.
For the orthogonal group, its Haar measure can be constructed as follows. First sample, 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 on the matrix; the resulting random variable takes values in and it is distributed according to the probability Haar measure on that group. Since the special orthogonal group is an open subgroup of the restriction of Haar measure of to gives a Haar measure on .
The same method as for can be used to construct the Haar measure on the unitary group. For the special unitary group , its Haar measure can be constructed as follows. First sample from the Haar measure on, and let , where may be any one of the angles, then independently sample from the uniform distribution on. Then is distributed as the Haar measure on.
Let be the set of all affine linear transformations of the form for some fixed with Associate with the operation of function composition, which turns into a non-abelian group. can be identified with the right half plane under which the group operation becomes A left-invariant Haar measure on is given by
and
for any Borel subset of This is because if is an open subset then for fixed, integration by substitution gives
while for fixed,
On any Lie group of dimension a left Haar measure can be associated with any non-zero left-invariant -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 , b > a > 0. For example, a = 1 and b = Euler’s number e yields and area equal to log = 1. Then for any positive real number c the area over the interval equals log 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 , 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.
If is the group of non-zero quaternions, then can be seen as an open subset of. A Haar measure is given by
where denotes the Lebesgue measure in and is a Borel subset of.
If is the additive group of -adic numbers for a prime, then a Haar measure is given by letting have measure, where is the ring of -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 with nonempty define to be the smallest number of left translates of that cover . This is not additive on compact sets, though it does have the property that for disjoint compact sets provided that is a sufficiently small open neighborhood of the identity. The idea of Haar measure is to take a sort of limit of as 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 with non-empty interior and for a compact set 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.
The function is additive on disjoint compact subsets of, which implies that it is a regular content. From a regular content one can construct a measure by first extending to open sets by inner regularity, then to all sets by outer regularity, and then restricting it to Borel sets.

A construction using compactly supported functions

Cartan introduced another way of constructing Haar measure as a Radon measure, which is similar to the construction above except that,, and are positive continuous functions of compact support rather than subsets of. In this case we define to be the infimum of numbers such that is less than the linear combination of left translates of for some.
As before we define
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 extends to a positive linear functional on compactly supported continuous functions and so gives a Haar measure.