© Instytut Matematyczny PAN, 2019 We study the invertible version of Furstenberg’s ‘ergodic CP shift systems’, which describe a random walk on measures on Euclidean space. These measures are by definition invariant under a scaling procedure, and satisfy a condition called adapt-edness under a ‘local’ translation operation. We show that the distribution is in fact non-singular with respect to a suitably defined translation operator on measures, and derive discrete and continuous pointwise ergodic theorems for the translation action.