continuous epimorphism of compact groups preserves Haar measure
Theorem - Let be compact Hausdorff topological groups. If is a continuous surjective homomorphism, then is a measure preserving transformation, in the sense that it preserves the normalized Haar measure.
: Let be the Haar measure in (normalized, i.e. ). Let be defined for measurable subsets of by
The inclusion is obvious. To prove the other inclusion notice that if then for some . Hence, , i.e . It now follows that .
Thus, equality (1) and the fact that is a Haar measure imply that
Since is surjective it follows that is right invariant. It is not difficult to see that is regular, finite on compact sets and . Hence, is the normalized Haar measure in and, by definition, we have that
Thus, preserves the Haar measure.
|Title||continuous epimorphism of compact groups preserves Haar measure|
|Date of creation||2013-03-22 17:59:06|
|Last modified on||2013-03-22 17:59:06|
|Last modified by||asteroid (17536)|