The boundary model for the continuous cohomology of Isom${}^{+}({\mathbb{H}}^n)$

We prove that the continuous cohomology of ${\text{Isom}}^{+}({\mathbb{H}}^n)$ can be measurably realized on the boundary of hyperbolic space. This implies in particular that for ${\text{Isom}}^{+}({\mathbb{H}}^n)$ the comparison map from continuous bounded cohomology to continuous cohomology is injective in degree $3$. We furthermore prove a stability result for the continuous bounded cohomology of $\text{Isom}({\mathbb{H}}^n)$ and $\text{Isom}({\mathbb{H}}_{\mathbb{C}}^n)$.