  In this paper, we investigate the existence of random absolutely continuous
invariant measures (ACIP) for random expanding on average Saussol maps in
higher dimensions. This is done by the establishment of a random Lasota-Yorke
inequality for the transfer operators on the space of bounded oscillation. We
prove that the number of ergodic skew product ACIPs is finite and provide an
upper bound for the number of these ergodic ACIPs. This work can be seen as a
generalization of the work in BGT on admissible random Jab{\l}o\'nski maps to a
more general class of higher dimensional random maps.
