# Haar integral

Let $\Gamma$ be a locally compact topological group and $\mathcal{C}$ be the algebra of all continuous real-valued functions on $\Gamma$ with compact support. In addition we define $\mathcal{C}^{+}$ to be the set of non-negative 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 non-negative 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 real-valued function $I^{+}$

1. 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. 2.

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

3. 3.

(Translation-Invariance). $I(f(\delta\gamma))=I(f(\gamma))$ 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 translation-invariance: $\int_{\gamma\in\Gamma}f(\gamma\delta)=\int_{\gamma\in\Gamma}f(\gamma)$ 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). Springer-Verlag, New York, 1988.
• HG Gochschild, G.: The Structure of Lie Groups. Holden-Day, San Francisco, 1965.
Title Haar integral HaarIntegral 2013-03-22 13:39:56 2013-03-22 13:39:56 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Definition msc 28C05 normalized Haar integral