Last updated: 27 November 2014
Let be a topological group, that is a group with a topology in which the operations multiplication and taking inverse are continuous.
We assume further is locally compact Hausdorff with respect to its topology.
A locally compact Hausdorff topological group has a left Ilaar measure. Left Haar measure is unique up to multiplication by a real positive constant.
Giving a left Haar measure is equivalent to giving an integral on functions functions on with compact support, which satisfies:
H1: | is a continuous linear functional. |
H2: |
(Positivity) for all taking non-negative real values. There is such an whose integral is non-zero. |
H3: |
(Left Invariance) for all integrable functions and all |
The three conditions above characterize left Haar integral up to a positive multiple.
The conditions H1 and H2 are related to the topology on
A function on is said to have compact support if it zero on the complement of a compact subset of or equivalently the closure of is compact. The set of such functions form a subspace of all functions on Such a function will be called positive if it is non-trivial and takes values in A linear functional on is a linear map from to A positive linear functional on is is non-trivial linear functional that takes values in for all positive We consider positive linear functionals which are continuous with respect to the topology on given by the sup-norm, (Setting gives a metric on
Measure theory shows every positive continuous linear functional is given by integration with respect to a unique regular Borel measure (A Borel measure is a measure on the generated by the open subsets of A Borel measure is called regular if compact sets have finite measure and for every of finite measure Given a Borel measure denotes the space of functions which are absolutely integral with respect to the measure. For we denote its integral by When the measure is regular is a linear subspace of Then defines a continuous positive linear functional on This sets up a one to one correspondence between regular Borel measures and positive continuous linear functionals on
The one to one correspondence between measure and positive continuous linear functionals arises from the locally compact topology on We now consider the group action.
Left action, on functions with domain is defined by setting and right is defined by Because is a topological group, if is open, then so too are both and for every Similarly for compact subsets and similarly too for all sets in
We deduce firstly that if is continuous of compact support then so too are both and for any Secondly we have actions on Borel measures taking regular measures to regular measures. Given a measure and we form measures and the first giving measure and the second giving measure Thus it makes sense to change variables in integrals as follows and similarly for right Thus we have
Exercise: If we define left on measures by and right on measures by then for
A Borel measure on is called left invariant if: (This is equivalent to for all A left invariant regular Borel measure is called a left Haar measure.
(1) | Left Haar measure is unique up to multiplication by a positive constant. |
(2) | The Haar measure of compact subsets of with open interior is non-zero. |
(3) | A Haar measure is determined by fixing the measure of any compact subset of with open interior. |
Having made a fixed a choice of left Haar measure we write We can express left invariance thus: This is equivalent, to H3 above. In terms of change of variable this becomes simply,
Example: Discrete Groups.
Every group is locally compact, with respect to the discrete topology. Counting measure on subsets, for is a left Haar measure on
The compact subsets are the finite subsets of For for all but finitely many and
A locally compact topological group also has a right Haar measures unique up to positive multiple. In general a left Haar measure is not a right Haar measure. The relationship between left Haar measure and right Haar measure is described by the modular function of which is defined below.
If is a left Haar measure on then given any so too is where for Hence these measures differ by a positive real multiple, which depends on This multiple does depend on not the choice of left Haar measure
The function defined by is called the modular function of
We can express this alternatively as Integration gives: Thus the modular function can alternatively be defined by
The modular function of is a continuous homomorphism from to
Exercise: Show the modular function is well defined and a homomorphism.
A Left Haar measure on a locally compact topological group is a right Haar measure if and only if the modular function of is trivial.
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Proof. |
This follows directly from the definition of the modular function. |
A locally compact topological group for which left Haar measures are right Haar measures, i.e. those whose modular function is trivial, are called unimodular.
Exercise/Examples
(1) | Groups with the discrete topology are unimodular. |
(2) | If is locally compact abelian then it is unimodular. |
(3) | If is compact then it is unimodular. |
If a left Haar measure on is given by then a tight Haar measure is given by
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Proof. |
Note this argument is entirely algebraic. |
There is a second way to relate left and right Haar measure. If is a left Haar measure then is a right Haar measure. In terms of integrals is defined by, In terms of measures, if then where Since and are right Haar measures they must differ by a positive multiple. In fact they are always equal and we have
In general is not equal to unless is unimodular.
Other Examples of Haar measures:
(1) | under addition. This has Haar measure giving the usual Riemann integral for a continuous function zero outside the closed bounded interval |
(2) | The group, of positive real numbers under multiplication. This has Haar measure |
(3) | invertible matrices with entries This has Haar measure This group is unimodular. |
(4) | The group of upper triangular in This has left Haar measure This group is not unimodular unless |