Topological group

Haar measure

Given a locally compact Hausdorff topological group , there exist nontrivial, regular, locally finite, left- and right-invariant Borel measures on , unique up to multiplication by a positive constant, called the left Haar measure and right Haar measure of respectively. group Invariance means given Borel set and a group element

Proof

proof Too advanced for now

The main use of the Haar measure is analogous to the Reärrangement lemma in finite contexts.

Further terminology

• A Unimodular group is a group for which .

Explicit constructions and examples

• Haar measure of a discrete group is counting measure
• Haar measure of a compact Lie group

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