The Lebesgue decomposition for lattices of projection operators (Q1217217)

In this note, we establish the Lebesgue decomposition for s-bounded vector valued additive functions defined on lattices of sets. Restricting measures to the sets in an algebra of sets corresponds to considering an algebra of projection operators; this identification gives the Lebesgue decomposition for s-bounded additive functions defined on an algebra of sets as a consequence of the decomposition with respect to an algebra of projection operators that the author established in [Duke Math. J. 30, 553--556 (1963; Zbl 0118.05101)]. The corresponding restrictions of set functions to elements of a lattice of sets correspond to a lattice of projection operators. Consequently, our decomposition with respect to lattices of projection operators yields the Lebesgue decomposition for s-bounded additive functions defined on a lattice of sets as a special case.