Haar Measures

  • Michel Simonnet
Part of the Universitext book series (UTX)


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.


Compact Subset Invariant Measure Homogeneous Space Compact Group Compact Space 
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Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Michel Simonnet
    • 1
  1. 1.Department of MathematicsUniversity of DakarSenegal

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