  Let $G, H$ be two Kleinian groups with homeomorphic quotients $\mathbb H^3/G$
and $\mathbb H^3/H$. We assume that $G$ is of divergence type, and consider the
Patterson-Sullivan measures of $G$ and $H$. The measurable rigidity theorem by
Sullivan and Tukia says that a measurable and essentially directly measurable
equivariant boundary map $\widehat k$ from the limit set $\Lambda_G$ of $G$ to
that of $H$ is either the restriction of a M\"{o}bius transformation or totally
singular. In this paper, we shall show that such $\widehat k$ always exists. In
fact, we shall construct $\widehat k$ concretely from the Cannon-Thurston maps
of $G$ and $H$.
