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continuous epimorphism of compact groups preserves Haar measure
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(Theorem)
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Theorem - Let $G, H$ be compact Hausdorff topological groups. If $\phi:G \longrightarrow H$ is a continuous surjective homomorphism, then $\phi$ is a measure preserving transformation, in the sense that it preserves the normalized Haar measure.
$\,$
Proof: 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
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
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(1) |
The inclusion $\supseteq$ is obvious. To prove the other inclusion notice that if $z \in \phi^{-1}(\phi(s)E)$ then $\phi(z) = \phi(s)t$ for some $t \in E$ . Hence, $\phi(s^{-1}z)=t$ , i.e $s^{-1}z \in \phi^{-1}(E)$ . It now follows that $z=s(s^{-1}z) \in s\phi^{-1}(E)$ .
Thus, equality (1) and the fact that $\mu$ is a Haar measure imply that
Since $\phi$ 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
Thus, $\phi$ preserves the Haar measure. $\square$
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Cross-references: compact sets, finite, regular, imply, equality, inclusion, translations, invariant, measure, easy to see, subsets, measurable, Haar measure, transformation, measure preserving, homomorphism, surjective, continuous, topological groups, Hausdorff, compact, theorem
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This is version 6 of continuous epimorphism of compact groups preserves Haar measure, born on 2008-04-10, modified 2009-01-02.
Object id is 10495, canonical name is ContinuousEpimorphismOfCompactGroupsPreservesHaarMeasure.
Accessed 728 times total.
Classification:
| AMS MSC: | 22C05 (Topological groups, Lie groups :: Compact groups) | | | 28C10 (Measure and integration :: Set functions and measures on spaces with additional structure :: Set functions and measures on topological groups, Haar measures, invariant measures) | | | 37A05 (Dynamical systems and ergodic theory :: Ergodic theory :: Measure-preserving transformations) |
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Pending Errata and Addenda
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