  We show that if G is a discrete subgroup of the group of the isometries of
the hyperbolic k-space H^k, and if R is a representation of G into the group of
the isometries of H^n, then any R-equivariant map F from H^k to H^n extends to
the boundary in a weak sense in the setting of Borel measures. As a consequence
of this fact, we obtain an extension of a result of Besson, Courtois and Gallot
about the existence of volume non-increasing, equivariant maps. Moreover, under
an additional hypothesis, we show that the weak extension we obtain is actually
a measurable R-equivariant map from the boundary of H^k to the closure of H^n.
We use this fact to obtain measurable versions of Cannon-Thurston-type results
for equivariant Peano curves.
