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 formula_1 be a locally compact Hausdorff topological group. The formula_2-algebra generated by all open subsets of formula_3 is called the Borel algebra. An element of the Borel algebra is called a Borel set. If formula_4 is an element of formula_3 and formula_6 is a subset of formula_3, then we define the left and right translates of formula_6 by "g" as follows:
Left and right translates map Borel sets onto Borel sets.
A measure formula_11 on the Borel subsets of formula_3 is called "left-translation-invariant" if for all Borel subsets formula_13 and all formula_14 one has
A measure formula_11 on the Borel subsets of formula_3 is called "right-translation-invariant" if for all Borel subsets formula_13 and all formula_14 one has
Haar's theorem.
There is, up to a positive multiplicative constant, a unique countably additive, nontrivial measure formula_11 on the Borel subsets of formula_3 satisfying the following properties:
Such a measure on formula_3 is called a "left Haar measure." It can be shown as a consequence of the above properties that formula_37 for every non-empty open subset formula_34. In particular, if formula_3 is compact then formula_40 is finite and positive, so we can uniquely specify a left Haar measure on formula_3 by adding the normalization condition formula_42.
In complete analogy, one can also prove the existence and uniqueness of a "right Haar measure" on formula_3. 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 formula_2-ring generated by compact sets, and defines Haar measures on these sets.
The left Haar measure satisfies the inner regularity condition for all formula_2-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 and a Haar measure on this group is not inner regular for the closed subset formula_46. (Compact subsets of this vertical segment are finite sets and points have measure formula_47, so the measure of any compact subset of this vertical segment is formula_47. 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.
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 formula_49 with formula_6 nonempty define formula_51 to be the smallest number of left translates of formula_6 that cover formula_53 (so this is a non-negative integer or infinity). This is not additive on compact sets formula_29, though it does have the property that formula_55 for disjoint compact sets formula_56 provided that formula_57 is a sufficiently small open neighborhood of the identity (depending on formula_58 and formula_59). The idea of Haar measure is to take a sort of limit of formula_60 as formula_57 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 formula_62 with non-empty interior (which exists as the group is locally compact) and for a compact set formula_58 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 formula_65 is additive on disjoint compact subsets of formula_3, which implies that it is a regular content. From a regular content one can construct a measure by first extending formula_65 to open sets by inner regularity, then to all sets by outer regularity, and then restricting it to Borel sets. (Even for open sets formula_57, the corresponding measure formula_69 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 formula_62, formula_58, and formula_57 are positive continuous functions of compact support rather than subsets of formula_3. In this case we define formula_60 to be the infimum of numbers formula_75 such that formula_76 is less than the linear combination formula_77 of left translates of formula_57 for some formula_79.
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 formula_65 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 formula_58, the individual terms formula_60 are not usually linear in formula_58.)
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 formula_85 on a compact group, one can find a convex combination formula_86 (where formula_87) of its left translates that differs from a constant function by at most some small number formula_88. Then one shows that as formula_88 tends to zero the values of these constant functions tend to a limit, which is called the mean value (or integral) of the function formula_85.
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 formula_91 satisfying the above regularity conditions and being finite on compact sets, but it need not coincide with the left-translation-invariant measure formula_11. 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 formula_11 and formula_94.
Indeed, for a Borel set formula_6, let us denote by formula_96 the set of inverses of elements of formula_6. If we define 
then this is a right Haar measure. To show right invariance, apply the definition:
Because the right measure is unique, it follows that formula_100 is a multiple of formula_94 and so
for all Borel sets formula_6, where formula_104 is some positive constant.
The modular function.
The "left" translate of a right Haar measure is a right Haar measure. More precisely, if formula_94 is a right Haar measure, then
is also right invariant. Thus, by uniqueness up to a constant scaling factor of the Haar measure, there exists a function formula_107 from the group to the positive reals, called the Haar modulus, modular function or modular character, such that for every Borel set formula_6
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 into the multiplicative group of positive real numbers. A group is called unimodular if the modular function is identically formula_110, 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 Lie group need not be unimodular.
In this group a left Haar measure is given by formula_112, and a right Haar measure by formula_113.
Measures on homogeneous spaces.
If the locally compact group formula_3 acts transitively on a homogeneous space formula_115, one can ask if this space has an invariant measure, or more generally a semi-invariant measure with the property that formula_116 for some character formula_117 of formula_3. A necessary and sufficient condition for the existence of such a measure is that the restriction formula_119 is equal to formula_120, where formula_107 and formula_122 are the modular functions of formula_3 and formula_124 respectively. 
In particular an invariant measure on formula_115 exists if and only if the modular function formula_107 of formula_3 restricted to formula_124 is the modular function formula_122 of formula_124.
Example.
If formula_3 is the group formula_132 and formula_124 is the subgroup of upper triangular matrices, then the modular function of formula_124 is nontrivial but the modular function of formula_3 is trivial. The quotient of these cannot be extended to any character of formula_3, so the quotient space formula_115 (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 formula_85 on formula_3. This integral is called the Haar integral and is denoted as:
where formula_11 is the Haar measure.
One property of a left Haar measure formula_11 is that, letting formula_143 be an element of formula_3, the following is valid:
for any Haar integrable function formula_85 on formula_3. 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 for compact groups by John von Neumann.
Unless formula_3 is a discrete group, it is impossible to define a countably additive left-invariant regular measure on "all" subsets of formula_3, 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. To prove the existence of a Haar measure on a locally compact group formula_3 it suffices to exhibit a left-invariant Radon measure on formula_3.
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.