Lie Groups, 2014. The modular function
The modular function for a locally compact Hausdorff group is defined by
ν(g-1S)=Δ(g)ν(S)
for all Borel sets S and all g in G,
where ν is a right Haar measure (according to
wikipedia).
Since μ(S)=ν(S-1) defines a left Haar measure,
it is equivalent to request
μ(Tg)=Δ(g)μ(T)
for all Borel sets T and a left Haar measure. This again is equivalent to requiring
that
∫ f(xg-1) dμ = Δ(g) ∫ f(x) dμ
for all integrable functions
(consider the indicator function of T)
For a Lie group it follows from Prop 19.3 that
∫ f(x) rg * (dx)=
∫ rg * (f(xg-1)dx)=
∫ f(xg-1) dx
and then by Lemma 19.12
∫ f(xg-1) dx =|detAd(g)-1|
∫ f(x) dx
Hence the proper definition of the modular function in this case is
Δ(g)=|detAd(g)-1|