TITLE: Invariance of the Haar measure -- upon inverses?
QUESTION [8 upvotes]: To simplify, let's assume $\mu$ is an invariant Haar measure on a commutative locally compact group $G$. Then, this means that $\mu$ is invariant under translation $\mu(U)=\mu(aU)$.
However, I noticed is that, for cases like $G=(\Bbb R, +)$, or $G=\Bbb R^*, \times)$, $\mu$, is also invariant, up to a minus, under inverses. Specifically:
$$\begin{aligned}x\mapsto -x&&x\mapsto x^{-1}\\
dx\mapsto -dx&&\frac {dx} x \mapsto -\frac {dx} x\end{aligned}$$
This is pretty neat when we are integrating over some domain, because after taking the inverse on the integration variable, we can also get a second minus sign by swapping the limits of integration. 
Does this generalize to other locally compact groups?

REPLY [2 votes]: It depends of the value of "generalize". See: 
http://en.wikipedia.org/wiki/Haar_measure#The_modular_function
https://mathoverflow.net/questions/130241/how-do-these-two-haar-measures-on-sl2-r-compare