Lebesgue decomposition and bartle—dunford—schwartz theorem in pseudo-D-lattices

Let L be a pseudo-D-lattice. We prove that the exhaustive lattice uniformities on L which makes the operations of L uniformly continuous form a Boolean algebra isomorphic to the centre of a suitable complete pseudo-D-lattice associated to L. As a consequence, we obtain decomposition theorems—such as Lebesgue and Hewitt—Yosida decompositions—and control theorems—such as Bartle—Dunford—Schwartz and Rybakov theorems—for modular measures on L.