# continuous epimorphism of compact groups preserves Haar measure

\PMlinkescapephrase

right \PMlinkescapephrasepreserves

Theorem - Let $G,H$ be compact^{} Hausdorff^{} topological groups^{}. If $\varphi :G\u27f6H$ is a continuous^{} surjective^{} homomorphism^{}, then $\varphi $ is a measure preserving transformation, in the sense that it preserves the normalized Haar measure.

$$

*:* Let $\mu $ be the Haar measure in $G$ (normalized, i.e. $\mu (G)=1$). Let $\nu $ be defined for measurable subsets $E$ of $H$ by

$$\nu (E)=\mu ({\varphi}^{-1}(E))$$ |

It is easy to see that $\nu $ defines a measure^{} in $H$. Let us now see that $\nu $ is invariant under right translations. For every $s\in G$ and every measurable subset $E\subset H$ we have that

${\varphi}^{-1}(\varphi (s)E)=s{\varphi}^{-1}(E)$ | (1) |

The inclusion $\supseteq $ is obvious. To prove the other inclusion notice that if $z\in {\varphi}^{-1}(\varphi (s)E)$ then $\varphi (z)=\varphi (s)t$ for some $t\in E$. Hence, $\varphi ({s}^{-1}z)=t$, i.e ${s}^{-1}z\in {\varphi}^{-1}(E)$. It now follows that $z=s({s}^{-1}z)\in s{\varphi}^{-1}(E)$.

Thus, equality (1) and the fact that $\mu $ is a Haar measure imply that

$$\nu (\varphi (s)E)=\mu \left({\varphi}^{-1}(\varphi (s)E)\right)=\mu (s{\varphi}^{-1}(E))=\mu ({\varphi}^{-1}(E))=\nu (E)$$ |

Since $\varphi $ is surjective it follows that $\nu $ is right invariant. It is not difficult to see that $\nu $ is regular^{}, finite on compact sets and $\nu (H)=1$. Hence, $\nu $ is the normalized Haar measure in $H$ and, by definition, we have that

$$\nu (E)=\mu ({\varphi}^{-1}(E))$$ |

Thus, $\varphi $ preserves the Haar measure. $\mathrm{\square}$

Title | continuous epimorphism of compact groups preserves Haar measure |
---|---|

Canonical name | ContinuousEpimorphismOfCompactGroupsPreservesHaarMeasure |

Date of creation | 2013-03-22 17:59:06 |

Last modified on | 2013-03-22 17:59:06 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 9 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 37A05 |

Classification | msc 28C10 |

Classification | msc 22C05 |