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The Lebesgue decomposition of measures on orthomodular posets. (English) Zbl 0617.46065
This is an investigation of the Lebesgue decomposition of positive measures on an orthomodular poset L. Conditions for subcones of positive measures on L are obtained under which the requirement of a positive Lebesgue decomposition is equivalent to L being a Boolean lattice. This result is used to give a measure-theoretic characterization of associative JBW-algebras amongst all JBW-algebras. Permanence properties of this decomposition are studied.

MSC:
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L70 Nonassociative selfadjoint operator algebras
28C99 Set functions and measures on spaces with additional structure
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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