Exercise 6, § 3

Let $P$ be the set of complex numbers $z = x + iy$ whose imaginary part is $> 0$. Show that $\mathrm{SL}(2, \mathbf{R})$ operates transitively and continuously on the left in $P$ by

and that $P$ thus becomes a topological homogeneous space for $\mathrm{SL}(2, \mathbf{R})$. Show that $y^{-2} dx dy$ is an invariant measure on $P$.