Let be a locally compact Hausdorff topological group and a left Haar measure. Although left and right Haar measures in always exist, they generally do not coincide, i.e. a left Haar measure is usually not invariant under right translations. Nevertheless, the right translations of a left Haar measure can be easily described as explained in the following theorem.
Moreover, if is an integrable function then
The function is called the modular function of (notice that, by uniqueness up to scalar multiple of left Haar measures, only depends on ). Other names for that can be found are: Haar modulus, or modular character or modular homomorphism.
We now prove the above theorem, except the continuity of (which is slightly harder to obtain).
Proof (except continuity of ):
Let . The function , defined on measurable subsets by
is easily seen to be a measure in . Moreover, is left invariant (since is left invariant) and satisfies the additional conditions to be a left Haar measure. By the uniqueness of left Haar measures, must be a multiple of , i.e. for some positive scalar . Thus, we have proven that for every measurable subset
Now for we have that , but also
So, we can see that, for every measurable subset ,
Hence, . Thus, is an homomorphism.
|Date of creation||2013-03-22 17:58:18|
|Last modified on||2013-03-22 17:58:18|
|Last modified by||asteroid (17536)|