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.

one has There is, up to a positive multiplicative constant, a unique countably additive, nontrivial measure

In complete analogy, one can also prove the existence and uniqueness of a right Haar measure on

(Compact subsets of this vertical segment are finite sets and points have measure

The existence and uniqueness (up to scaling) of a left Haar measure was first proven in full generality by André Weil.

[6] The special case of invariant measure for second-countable locally compact groups had been shown by Haar in 1933.

[1] 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 in invariant under multiplication by positive real numbers.

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.

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 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.)

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

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.

Von Neumann gave a method of constructing Haar measure using mean values of functions, though it only works for compact groups.

) of its left translates that differs from a constant function by at most some small number

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.

It can also be proved that there exists a unique (up to multiplication by a positive constant) right-translation-invariant Borel measure

satisfying the above regularity conditions and being finite on compact sets, but it need not coincide with the left-translation-invariant measure

is a right Haar measure, then for any fixed choice of a group element g, is also right invariant.

Thus, by uniqueness up to a constant scaling factor of the Haar measure, there exists a function

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.

This example shows that a solvable Lie group need not be unimodular.

This is immediate for indicator functions: which is essentially the definition of left invariance.

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 subsets of

The Haar measures are used in harmonic analysis on locally compact groups, particularly in the theory of Pontryagin duality.

[10][11][12] To prove the existence of a Haar measure on a locally compact group

For example, a right Haar measure for a family of distributions with a location parameter results in the Pitman estimator, which is best equivariant.

[15] 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,[3] 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.