Let be a locally compact topological group and be the algebra of all continuous real-valued functions on with compact support. In addition we define to be the set of non-negative functions that belong to . The Haar integral is a real linear map of into the field of the real number for if it satisfies:
is not the zero map
only takes non-negative values on
has the following property for all elements of and all element of .
The Haar integral may be denoted in the following way (there are also other ways):
or or or
The following are necessary and sufficient conditions for the existence of a unique Haar integral: There is a real-valued function
(Linearity). where and .
(Positivity). If for all then .
(Translation-Invariance). for any fixed and every in .
An additional property is if is a compact group then the Haar integral has right translation-invariance: for any fixed .
In addition we can define normalized Haar integral to be since is compact, it implies that is finite.
(The proof for existence and uniqueness of the Haar integral is presented in [HG] on page 9.)
(the information of this entry is in part quoted and paraphrased from [GSS])
- GSS Golubsitsky, Martin. Stewart, Ian. Schaeffer, G. David.: Singularities and Groups in Bifurcation Theory (Volume II). Springer-Verlag, New York, 1988.
- HG Gochschild, G.: The Structure of Lie Groups. Holden-Day, San Francisco, 1965.