Measures and Probabilities pp 425-464 | Cite as

# 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 **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.

### Keywords

Convolution Radon Sine diAl## Preview

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