Haar measure of a compact Lie group

Let be an -dimensional Compact Lie group, be its corresponding Lie algebra¹ with basis , and be a coördinate chart with . For each define so that

where both sides are clearly elements of . Then for any Borel set , the unique left and right Haar measure is given by #m/thm/group

where is a normalisation constant. The Haar measure is defined for the whole of by translating the chart (enabled by invariance).

Proof this is a Haar measure

Let define a chart with , and let be a transition map such that for all . Then

i.e. and thus

as required.

For we define the chart

then

which gives left-invariance.

For each let so that . Letting , i.e. , then

i.e. . But

and since is compact all integrals are finite, thus the , i.e. is unimodular. Therefore is right-invariant.

Proof of uniqueness

Let both be two sided Haar measures normalised such that . Then for any

thus .

Properties

Let and be a bijection (e.g. taking the inverse of each group element)

Proof of property 3

From properties 1 and 2

where we used the normalisation of the group to 1.

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