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Haar integral
Let $\Gamma$ be a locally compact topological group and $\mathcal{C}$ be the algebra of all continuous realvalued functions on $\Gamma$ 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 $\Gamma$ 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 $\Gamma$.
The Haar integral may be denoted in the following way (there are also other ways):
$\int_{{\gamma\in\Gamma}}f(\gamma)$ or $\int_{\Gamma}f$ or $\int_{\Gamma}fd\gamma$ 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 $f(\gamma)\geq 0$ for all $\gamma\in\Gamma$ then $I^{+}(f(\gamma))\geq 0$.
3. (TranslationInvariance). $I(f(\delta\gamma))=I(f(\gamma))$ for any fixed $\delta\in\Gamma$ and every $f$ in $\mathcal{C}^{+}$.
An additional property is if $\Gamma$ is a compact group then the Haar integral has right translationinvariance: $\int_{{\gamma\in\Gamma}}f(\gamma\delta)=\int_{{\gamma\in\Gamma}}f(\gamma)$ for any fixed $\delta\in\Gamma$.
In addition we can define normalized Haar integral to be $\int_{\Gamma}1=1$ since $\Gamma$ is compact, it implies that $\int_{\Gamma}1$ 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.
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