Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. One can put a (left) Haar measure $\mu$ on $G$ and a Lebesgue measure $\lambda$ on $\mathfrak{g}$ which are both unique up to constants.
My question is whether fixing $\mu$ on $G$ picks out $\lambda$ uniquely on $\mathfrak{g}$ and conversely.
I think a natural first step in answering this question is to consider the exponential map $\exp:\mathfrak{g}\to G$ and try to use the change of variables formula. However there are two issues that one faces:
- The exponential map is in general a local diffeomorphism only.
- The Lebesgue measure transferred from $\mathfrak{g}$ to $G$ via the change of variables formula need not be a Haar measure: $$\int_G f(g)\, d\mu(g):=\int_{\mathfrak{g}}f(\exp X) |\mathrm{Jac}\exp|\, d\lambda(X) $$
Can someone please shed some light on the relation between the measures on the Lie group and the Lie algebra and resolving the above two issues? (I think the first issue is not severe, as $\exp$ is a local diffeomorphism and that should be enough for transferring the measure from one side to the other, if at all possible for a group $G$.)
Thanks in advance for any comments/answers.