modular function

Let G be a locally compact Hausdorff topological groupMathworldPlanetmath and μ 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 theoremMathworldPlanetmath.

Theorem - Let G be a locally compact Hausdorff topological group and μ a left Haar measure in G. Then, there exists a continuousPlanetmathPlanetmath homomorphismPlanetmathPlanetmathPlanetmathPlanetmath Δ:G+ such that, for every tG and every measurable subset A


Moreover, if f:G is an integrable function then


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

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

Proof (except continuity of Δ):

Let tG. The function ν, defined on measurable subsets A by


is easily seen to be a measureMathworldPlanetmath in G. Moreover, ν is left invariant (since μ is left invariant) and satisfies the additional conditions to be a left Haar measure. By the uniqueness of left Haar measures, μ must be a multiple of ν, i.e. μ=Δ(t)ν for some positive scalar Δ(t)+. Thus, we have proven that for every measurable subset A


Now for s,tG we have that μ(Ast)=Δ(st)-1μ(A), but also

  • μ(Ast)=Δ(t)-1μ(As), and

  • μ(As)=Δ(s)-1μ(A)

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


Hence, Δ(st)=Δ(s)Δ(t). Thus, Δ is an homomorphism.

The statement about integrals of functions follows easily by approximation by simple functionsMathworldPlanetmath. For simple functions it is easy to see it is true using the now established condition μ(At)=Δ(t-1)μ(A).

Title modular function
Canonical name ModularFunction
Date of creation 2013-03-22 17:58:18
Last modified on 2013-03-22 17:58:18
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 8
Author asteroid (17536)
Entry type Definition
Classification msc 22D05
Classification msc 28C10
Synonym Haar modulus
Synonym modular character
Synonym modular homomorphism