Haar integral

Let Γ be a locally compact topological group and 𝒞 be the algebraPlanetmathPlanetmath of all continuous real-valued functions on Γ with compact support. In addition we define 𝒞+ to be the set of non-negative functions that belong to 𝒞. The Haar integral is a real linear map I of 𝒞 into the field of the real number for Γ if it satisfies:

  • I is not the zero map

  • I only takes non-negative values on 𝒞+

  • I has the following property I(γf)=I(f) for all elements f of 𝒞 and all element γ of Γ.

The Haar integral may be denoted in the following way (there are also other ways):

γΓf(γ) or Γf or Γf𝑑γ or I(f)

The following are necessary and sufficient conditions for the existence of a unique Haar integral: There is a real-valued function I+

  1. 1.

    (Linearity).I+(λf+μg)=λI+(f)+μI+(g) where f,g𝒞+ and λ,μ+.

  2. 2.

    (Positivity). If f(γ)0 for all γΓ then I+(f(γ))0.

  3. 3.

    (Translation-Invariance). I(f(δγ))=I(f(γ)) for any fixed δΓ and every f in 𝒞+.

An additional property is if Γ is a compact group then the Haar integral has right translation-invariance: γΓf(γδ)=γΓf(γ) for any fixed δΓ. In addition we can define normalized Haar integral to be Γ1=1 since Γ is compactPlanetmathPlanetmath, it implies that Γ1 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 GroupsMathworldPlanetmath. Holden-Day, San Francisco, 1965.
Title Haar integral
Canonical name HaarIntegral
Date of creation 2013-03-22 13:39:56
Last modified on 2013-03-22 13:39:56
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Definition
Classification msc 28C05
Defines normalized Haar integral