Let be a locally compact group.
Left Haar measure
A left Haar measure is a left-translation-invariant countably additive regular nontrivial measure on the Borel subsets of . The conditions are explained below:
|Borel subset||A subset in the -algebra generated by open subsets of|
|Left-translation-invariant measure||If is measurable, and , then is measurable, and , where|
|Countably additive (part of the usual definition of measure)||If are all measurable sets that are pairwise disjoint, and their union is , then is the sum of the values .|
|Regular Borel measure||is finite for any compact subset . Also, for every Borel subset , and|
|Nontrivial measure||for any nonempty open subset of|
The left Haar measure for a locally compact group is unique up to scalar multiples, i.e., the quotient of any two left Haar measures is a scalar.
For a compact group, there is a unique choice of normalized Haar measure, i.e., a unique left Haar measure where the total measure of the group is .
Right Haar measure
A right Haar measure is a left-translation-invariant countably additive regular measure on the Borel subsets of .
The right Haar measure for a locally compact group is unique up to scalar multiples, i.e., the quotient of any two left Haar measures is a scalar.
Relationship between left and right Haar measure
Any left Haar measure on a group can be used to canonically define a right Haar measure: for a left Haar measure , define a correpsonding right Haar measure as:
This definition makes sense because is Borel if and only if is. The left Haar measure conditions on give rise to the right Haar measures on .
For a compact group, the following additional features are true:
- The left Haar measures coincide with the right Haar measures. In particular, we just talk of a Haar measure without talking of left or right. This Haar measure thus satisfies the additional condition that for any measurable subset .
- Further, since the total measure of the group is finite, there is a natural choice of normalized Haar measure where the total measure of the group is .
Bi-invariant Haar measures?
A bi-invariant Haar measure is a measure that is both a left and a right Haar measure. Not all locally compact groups have bi-invariant Haar measures, but compact groups, locally compact abelian groups, and some other examples of locally compact groups have bi-invariant Haar measures. If a group has a bi-invariant Haar measure, then every left Haar measure is a right Haar measure, and hence a bi-invariant Haar measure.