We develop a pluripotential theory for random iteration on ${\mathbf P}^k$. We show the existence of a positive closed $(1,1)$ current and a measure on ${\mathbf P}^k$ which are invariant, in a certain sense, and which attract all positive closed $(1,1)$ currents and all measures, respectively, under normalized pull-back and averaging by the maps. Thus the concept of an exceptional set disappears as soon as we allow a slight amount of randomness in our system. We also consider the problem of push-forward of measures, and describe certain limiting measures in this case also, supported near the attractors for the perturbed map.