Compact Group - Haar Measure

Haar Measure

Compact groups all carry a Haar measure, which will be invariant by both left and right translation (the modulus function must be a continuous homomorphism to the positive multiplicative reals, and so 1). In other words these groups are unimodular. Haar measure is easily normalised to be a probability measure, analogous to dθ/2π on the circle.

Such a Haar measure is in many cases easy to compute; for example for orthogonal groups it was known to Hurwitz, and in the Lie group cases can always be given by an invariant differential form. In the profinite case there are many subgroups of finite index, and Haar measure of a coset will be the reciprocal of the index. Therefore integrals are often computable quite directly, a fact applied constantly in number theory.