Haar Measures

Summary

23.1 In this section we define invariant, relatively invariant, and quasi-invariant measures on a locally compact group.

23.2 On every locally compact group there exists a left-invariant measure. This measure is unique, up to a multiplicative constant (Theorem 23.2.1).

23.3 We define the modular function on a locally compact group.

23.4 Quasi-invariant measures on a locally compact group are all equivalent (Proposition 23.4.3).

23.5 Let G be a locally compact group and H a closed subgroup of G. With every measure λ on the homogeneous space G/H we can associate a measure λ^# on G (Theorem 23.5.2) that possesses interesting properties.

23.6 We study integration with respect to λ^# (Theorem 23.6.1 and Proposition 23.6.1).

23.7 We reconstitute λ from λ^# (Proposition 23.7.4).

23.8 There is only one class of quasi-invariant measures on G/H (Theorems 23.8.1 and 23.8.2).

23.9 In contrast, invariant measures exist on G/H only if the modular functions Δ _G and Δ _H of G and H coincide on H (Theorem 23.9.2).

23.10 By means of Euler angles we describe the invariant measure on the group SO(n + 1, R) of rotations of Rⁿ⁺¹ (Proposition 23.10.3).

23.11 We describe Haar measure on the group SH(n + 1, R) of hyperbolic rotations of Rⁿ⁺¹.

In this chapter and the following one, all locally compact spaces will be taken to be Hausdorff, unless otherwise stated.