Haar Measure - Examples

Examples

• A Haar measure on the topological group (R, +) which takes the value 1 on the interval is equal to the restriction of Lebesgue measure to the Borel subsets of R. This can be generalized to (Rn, +).

• If G is the group of nonzero real numbers with multiplication as operation, then a Haar measure μ is given by

for any Borel subset S of the nonzero reals.

This generalizes to the following:

• For G = GL(n,R), any left Haar measure is a right Haar measure and one such measure μ is given by

where dX denotes the Lebesgue measure on, the set of all -matrices. This follows from the change of variables formula.

• More generally, on any Lie group of dimension d a left Haar measure can be associated with any non-zero left-invariant d-form ω, as the Lebesgue measure |ω|; and similarly for right Haar measures. This means also that the modular function can be computed, as the absolute value of the determinant of the adjoint representation.

• In order to define a Haar measure μ on the unit circle T, consider the function f from onto T defined by f(t) = (cos(t),sin(t)). Then μ can be defined by

where m is the Lebesgue measure. The factor (2π)−1 is chosen so that μ(T) = 1.

• If G is the group of non-null quaternions, then G can be seen as an open subset of R4. A Haar measure μ is given by

where dx dy dz dw denotes the Lebesgue measure in R4 and S is a Borel subset of G.