Haar integral
Let $\mathrm{\Gamma}$ be a locally compact topological group and $\mathcal{C}$ be the algebra^{} of all continuous realvalued functions on $\mathrm{\Gamma}$ with compact support. In addition we define ${\mathcal{C}}^{+}$ to be the set of nonnegative functions that belong to $\mathcal{C}$. The Haar integral is a real linear map $I$ of $\mathcal{C}$ into the field of the real number for $\mathrm{\Gamma}$ if it satisfies:

•
$I$ is not the zero map

•
$I$ only takes nonnegative values on ${\mathcal{C}}^{+}$

•
$I$ has the following property $I(\gamma \cdot f)=I(f)$ for all elements $f$ of $\mathcal{C}$ and all element $\gamma $ of $\mathrm{\Gamma}$.
The Haar integral may be denoted in the following way (there are also other ways):
${\int}_{\gamma \in \mathrm{\Gamma}}f(\gamma )$ or ${\int}_{\mathrm{\Gamma}}f$ or ${\int}_{\mathrm{\Gamma}}f\mathit{d}\gamma $ or $I(f)$
The following are necessary and sufficient conditions for the existence of a unique Haar integral: There is a realvalued function ${I}^{+}$

1.
(Linearity).${I}^{+}(\lambda f+\mu g)=\lambda {I}^{+}(f)+\mu {I}^{+}(g)$ where $f,g\in {\mathcal{C}}^{+}$ and $\lambda ,\mu \in {\mathbb{R}}_{+}$.

2.
(Positivity). If $f(\gamma )\ge 0$ for all $\gamma \in \mathrm{\Gamma}$ then ${I}^{+}(f(\gamma ))\ge 0$.

3.
(TranslationInvariance). $I(f(\delta \gamma ))=I(f(\gamma ))$ for any fixed $\delta \in \mathrm{\Gamma}$ and every $f$ in ${\mathcal{C}}^{+}$.
An additional property is if $\mathrm{\Gamma}$ is a compact group then the Haar integral has right translationinvariance: ${\int}_{\gamma \in \mathrm{\Gamma}}f(\gamma \delta )={\int}_{\gamma \in \mathrm{\Gamma}}f(\gamma )$ for any fixed $\delta \in \mathrm{\Gamma}$.
In addition we can define normalized Haar integral to be ${\int}_{\mathrm{\Gamma}}1=1$ since $\mathrm{\Gamma}$ is compact^{}, it implies that ${\int}_{\mathrm{\Gamma}}1$ 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])
References
 GSS Golubsitsky, Martin. Stewart, Ian. Schaeffer, G. David.: Singularities and Groups in Bifurcation Theory (Volume II). SpringerVerlag, New York, 1988.
 HG Gochschild, G.: The Structure of Lie Groups^{}. HoldenDay, San Francisco, 1965.
Title  Haar integral 

Canonical name  HaarIntegral 
Date of creation  20130322 13:39:56 
Last modified on  20130322 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 