Haar measure

The Haar measure was introduced by Alfréd hair in mathematics in order to make the results of measure theory in group theory applicable.

It is a generalization of the Lebesgue measure. The Lebesgue measure is a measure on the Euclidean space, which is invariant under translations. The Euclidean space is a locally compact topological group with respect to addition. The Haar measure is defined for any locally compact (the always presupposed as Hausdorff ) topological group, especially so for any Lie group. Locally Compact groups with their Haar measures are examined in the harmonic analysis.

Definition

A (left ) Hair cal measure of a locally compact group G is a left invariant regular Borel measure which is positive on non-empty open subsets.

A measure μ is called here linksinvariant if for every Borel set A and each group element g

Or in integral notation

F and group elements g is valid for integrable functions.

If one replaces " linksinvariant " by the analog term " rechtsinvariant ", you get the concept of the right hair - measure. The left and the right Hair measure exist in any locally compact topological group and on each of them are uniquely determined up to a factor. If they match, then the group is called unimodular both. Abelian ( locally compact ) groups and compact groups are unimodular.

Proof of the existence

According to a version of the representation theorem of Riesz, it is sufficient to show the existence of a continuous, positive, left-invariant linear functional on the non-negative, real-valued, continuous functions with compact support on a locally compact group. In the real case is an example of one such, the Riemann integral, which can be continued to the Lebesgue integral, and thus induces the Lebesgue measure. The proof of existence is non- constructively possible over the set of Tychonoff.

To this end, we first define for each pair of non-negative, continuous functions with compact support with the covering number as

Denote the left shift, ie. Forever " finer " the overlap is then always " accurate ", but the coverage number is usually increasing. This can be get through the normalization in the handle, one defines

For any non-zero. However, this is generally still functional nonlinear - it is indeed homogeneous, but generally only subadditive and not additive. Crucial for the further proof is then the following inequality:

Consider now the neighborhood filter of the neutral element in and form the image filter under the picture that each assigns to the set of all for which the support of is included. Thus a filter is obtained thanks to the estimate in space and this space is compact by the theorem of Tychonoff. The filter thus has a contact point, is expected after that such a contact point has all the desired properties, in particular linearly, that is, a left hair integral.

Properties

The Haar measure of a locally compact topological group is just finally, if the group is compact. This fact makes it possible to perform averaging over infinite compact groups by integration with respect to this metric. One consequence, for example, that every finite-dimensional complex representation of a compact group is unitary with respect to a suitable scalar product. A singleton has exactly then a Hair measure non-zero if the group is discrete.

Examples

• The Lebesgue measure on and is the Haar measure on the additive groups respectively.
• Be the circle group, ie the compact group of complex numbers of magnitude 1 with the usual multiplication of complex numbers as a link. Refers to the Lebesgue measure on the interval and the function, the Haar measure is given by the size, that is, for each Borel set.
• The general linear array, the Haar measure is given by where the Lebesgue measure on.
• For a discrete group is the counting hair measure cal.
• The Haar measure on the multiplicative group is represented by the formula, wherein the Lebesgue measure is.

The modular function

Μ is a ( left invariant ) cal hair measure then also the mapping, where g is a fixed group unit. Because of the uniqueness of the Haar measure exists a positive real number Δ ( g ), so that

Δ is a continuous homomorphism of the group into the multiplicative group of positive real numbers, the modular function is called. Δ measures how much a (left ) Hair measure is also rechtsinvariant; and a group is unimodular if and only if its modular function is constant 1.