Haar measure

Given a locally compact Hausdorff topological group $𝐺$ , there exist nontrivial, regular, locally finite, left- and right-invariant Borel measures on $𝐺$ , unique up to multiplication by a positive constant, called the left Haar measure $𝜇 𝐿$ and right Haar measure $𝜇 𝑅$ of $𝐺$ respectively. #m/thm/group Invariance means given Borel set $𝑈 \subseteq 𝐺$ and a group element $𝑔 \in 𝐺$

$𝜇 𝐿 (𝑈) = 𝜇 𝐿 (𝑔 𝑈)$
$𝜇 𝑅 (𝑈) = 𝜇 𝑅 (𝑈 𝑔)$

Proof

#missing/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 $𝜇 𝐿 = 𝜇 𝑅 = 𝜇$ .

Explicit constructions and examples

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

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