modular function

Let $G$ be a locally compact Hausdorff topological group and $\mu$ a left Haar measure. Although left and right Haar measures in $G$ 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.

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Theorem - Let $G$ be a locally compact Hausdorff topological group and $\mu$ a left Haar measure in $G$. Then, there exists a continuous homomorphism $\Delta:G\longrightarrow\mathbb{R}^{+}$ such that, for every $t\in G$ and every measurable subset $A$

 $\mu(At)=\Delta(t^{-1})\mu(A)$

Moreover, if $f:G\longrightarrow\mathbb{C}$ is an integrable function then

 $\Delta(t)\int_{G}f(st)\mu(s)=\int_{G}f(s)\mu(s)$

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The function $\Delta$ is called the modular function of $G$ (notice that, by uniqueness up to scalar multiple of left Haar measures, $\Delta$ only depends on $G$). Other names for $\Delta$ that can be found are: Haar modulus, or modular character or modular homomorphism.

We now prove the above theorem, except the continuity of $\Delta$ (which is slightly harder to obtain).

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Proof (except continuity of $\Delta$):

Let $t\in G$. The function $\nu$, defined on measurable subsets $A$ by

 $\nu(A):=\mu(At)$

is easily seen to be a measure in $G$. Moreover, $\nu$ is left invariant (since $\mu$ is left invariant) and satisfies the additional conditions to be a left Haar measure. By the uniqueness of left Haar measures, $\mu$ must be a multiple of $\nu$, i.e. $\mu=\Delta(t)\nu$ for some positive scalar $\Delta(t)\in\mathbb{R}^{+}$. Thus, we have proven that for every measurable subset $A$

 $\mu(At)=\Delta(t)^{-1}\mu(A)$

Now for $s,t\in G$ we have that $\mu(Ast)=\Delta(st)^{-1}\mu(A)$, but also

• $\mu(Ast)=\Delta(t)^{-1}\mu(As)$, and

• $\mu(As)=\Delta(s)^{-1}\mu(A)$

So, we can see that, for every measurable subset $A$,

 $\Delta(st)^{-1}\mu(A)=\Delta(t)^{-1}\Delta(s)^{-1}\mu(A)$

Hence, $\Delta(st)=\Delta(s)\Delta(t)$. Thus, $\Delta$ is an homomorphism.

The statement about integrals of functions follows easily by approximation by simple functions. For simple functions it is easy to see it is true using the now established condition $\mu(At)=\Delta(t^{-1})\mu(A)$. $\square$

Title modular function ModularFunction 2013-03-22 17:58:18 2013-03-22 17:58:18 asteroid (17536) asteroid (17536) 8 asteroid (17536) Definition msc 22D05 msc 28C10 Haar modulus modular character modular homomorphism