Haar integral

Let $\mathrm{\Gamma }$ be a locally compact topological group and $𝒞$ be the algebra of all continuous real-valued functions on $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ if it satisfies:

• $I$ is not the zero map

• $I$ only takes non-negative values on ${𝒞}^{+}$

• $I$ has the following property $I(\gamma \cdot f)=I(f)$ for all elements $f$ of $𝒞$ 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 ${\int }_{\mathrm{\Gamma }}f𝑑\gamma$ or $I(f)$

The following are necessary and sufficient conditions for the existence ofa unique Haar integral:There is a real-valued function $I^{+}$

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

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

3. (Translation-Invariance). $I(f(\delta \gamma ))=I(f(\gamma ))$ for any fixed $\delta \in \mathrm{\Gamma }$ and every $f$ in ${𝒞}^{+}$.

An additional property is if $\mathrm{\Gamma }$ is a compact group then the Haar integral has right translation-invariance: ${\int }_{\gamma \in \mathrm{\Gamma }}f(\gamma \delta )={\int }_{\gamma \in \mathrm{\Gamma }}f(\gamma )$ 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])