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The Lebesgue decomposition theorem and the Nikodym convergence theorem on an orthomodular poset. (English) Zbl 0747.28004
Several classical measure theoretic results are established in the setting of additive functions on orthomodular posets with values in algebraic semigroups. An orthomodular poset (OMP) \(L\) is a generalization of a Boolean algebra in which existence of infima and suprema and distributivity of these lattice operations are not assumed. Throughout, the codomain of additive functions on \(L\) is either a semigroup or a group; when a topology is called for it is assumed to be a uniform topology. The paper begins with a nice introduction into the basic principles of OMP’s including the concepts of commuting pairs of elements and central elements, ideas which are important in dealing with the lack of infima and suprema, before attacking the principle results: (1) a Lebesgue decomposition theorem for two additive functions from an OMP into commutative semigroups; (2) a Lebesgue decomposition theorem where \(L\) is a \(\sigma\)-complete Boolean algebra and the codomain is a Hausdorff commutative topological group; (3) a generalization of the Nikodým convergence theorem for a sequence of strongly and countably additive functions from an OMP which is \(\sigma\)-orthocomplete (closed under suprema of countable collections of pairwise orthogonal elements) into a Hausdorff semigroup. In addition, while the Vitali-Hahn-Saks and Nikodým boundedness theorems are known to hold for strongly additive functions mapping \(\sigma\)-complete Boolean algebras into Hausdorff semigroups and Hausdorff commutative topological groups, respectively, examples are given to show that these two results fail when the domain is an OMP, even when it is assumed to be \(\sigma\)-orthocomplete.

28B10 Group- or semigroup-valued set functions, measures and integrals
06F99 Ordered structures
28A33 Spaces of measures, convergence of measures
28A60 Measures on Boolean rings, measure algebras