I am reading about geodesic currents on surfaces, which are measures on the space of geodesic curves on the surface.
The authors use a result that since a particular measure is invariant under a group action, it is unique up to a scalar. However I cannot find any such result in the literature. Does this result exist?
Here's a specific example, from "Hyperbolic structures on surfaces and geodesic currents" by Aramayona-Leiniger, pg 29:
"Even without the specific expression for L used in the proof, it is easy to see that $L(E_\delta)$ is a fixed constant multiple of \delta. For this, note that since L is invariant under the action of the entire isometry group of $\mathbb{H}^2$, $\delta\to L(E_\delta)$ defines a measure on any geodesic, and this measure is invariant under any isometry. Therefore, it is a constant multiple of the Lebesgue (length) measure. Because PSL$(2,\mathbb{R})$ acts transitively on the set of geodesics, the constant is independent of the geodesic."
I don't understand why we can say it is some multiple of the Lesbesgue measure. Any help would be greatly appreciated.
Edit: Thanks for the feedback!