  Let $\Gamma$ be a discrete group acting by isometries on a product
$X=X_1\times X_2$ of Hadamard spaces. We further require that $X_1$, $X_2$ are
locally compact and $\Gamma$ contains two elements projecting to a pair of
independent rank one isometries in each factor. Apart from discrete groups
acting by isometries on a product of CAT(-1)-spaces, the probably most
interesting examples of such groups are Kac-Moody groups over finite fields
acting on the Davis complex of their associated twin building. In a previous
article we showed that the regular geometric limit set $\Lim$ splits as a
product $F_\Gamma\times P_\Gamma$, where $F_\Gamma\subseteq\rand_1\times
\rand_2$ is the projection of the geometric limit set to $\rand_1\times
\rand_2$, and $P_\Gamma$ encodes the ratios of the speed of divergence of orbit
points in each factor. Our aim in this paper is a description of the limit set
from a measure theoretical point of view. We first study the conformal density
obtained from the classical Patterson-Sullivan construction, then generalize
this construction to obtain measures supported in each $\Gamma$-invariant
subset of the regular limit set and investigate their properties. Finally we
show that the Hausdorff dimension of the radial limit set in each
$\Gamma$-invariant subset of $\Lim$ is bounded above by the exponential growth
rate introduced in the previous article.
