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 R n +1 (Proposition 23.10.3).
23.11 We describe Haar measure on the group SH(n + 1, R) of hyperbolic rotations of R n +1.
In this chapter and the following one, all locally compact spaces will be taken to be Hausdorff, unless otherwise stated.
KeywordsCompact Subset Invariant Measure Homogeneous Space Compact Group Compact Space
Unable to display preview. Download preview PDF.