Haar Measure - The Modular Function

The Modular Function

The left translate of a right Haar measure is a right Haar measure. More precisely, if ν is a right Haar measure, then

is also right invariant. Thus, by uniqueness of the Haar measure, there exists a function Δ from the group to the positive reals, called the Haar modulus, modular function or modular character, such that for every Borel set S

Since right Haar measure is well-defined up to a positive scaling factor, this equation shows the modular function is independent of the choice of right Haar measure in the above equation.

The modular function is a continuous group homomorphism into the multiplicative group of nonzero real numbers. A group is unimodular if and only if the modular function is identically 1, or, equivalently, if the Haar measure is both left and right invariant. Examples of unimodular groups are abelian groups, compact groups, discrete groups (e.g., finite groups), semisimple Lie groups and connected nilpotent Lie groups. An example of a non-unimodular group is the group of affine transformations

on the real line. This example shows that a solvable Lie group need not be unimodular.