# compact groups are unimodular

Theorem - If $G$ is a compact Hausdorff topological group^{}, then $G$ is unimodular, i.e. it’s left and right Haar measures coincide.

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*Proof*:

Let $\mathrm{\Delta}$ denote the modular function^{} of $G$. It is enough to prove that $\mathrm{\Delta}$ is constant and equal to $1$, since this proves that every left Haar measure is right invariant.

Since $\mathrm{\Delta}$ is continuous^{} and $G$ is compact, $\mathrm{\Delta}(G)$ is a compact subset of ${\mathbb{R}}^{+}$. In particular, $\mathrm{\Delta}(G)$ is a bounded subset of ${\mathbb{R}}^{+}$.

But if $\mathrm{\Delta}$ is not identically one, then there is a $t\in G$ such that $\mathrm{\Delta}(t)>1$ (recall that $\mathrm{\Delta}$ is an homomorphism^{}). Hence, $\mathrm{\Delta}({t}^{n})=\mathrm{\Delta}{(t)}^{n}\u27f6\mathrm{\infty}$ as $n\in \mathbb{N}$ increases, which is a contradiction^{} since $\mathrm{\Delta}(G)$ is bounded. $\mathrm{\square}$

Title | compact groups are unimodular |
---|---|

Canonical name | CompactGroupsAreUnimodular |

Date of creation | 2013-03-22 17:58:23 |

Last modified on | 2013-03-22 17:58:23 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 4 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 22C05 |

Classification | msc 28C10 |