An Aumann-Shapley type operator in Pseudo-D-lattices

Let L be a sigma-complete pseudo-D-lattice and let BV be the Banach space of all real-valued, vanishing at zero, functions of bounded variation on L endowed with the variation norm. We prove the existence of a continuous Aumann-Shapley type operator s phi on the closed subspace of BV spanned by powers of nonatomic sigma-additive positive modular measures on L. Moreover we give an integral representation of phi on a class of functions that correspond to measure games.