Orthoalgebras and Noncommutative Measure Theory.

Notions of orthosummability and sigma -orthosummability are defined for orthoalgebras. It is shown that these notions extend the known notions of orthocompleteness and sigma-orthocompleteness for orthomodular posets and that the class of orthosummable (respectively, sigma-orthosummable) orthoalgebras properly contains the class of orthocomplete (respectively, sigma-orthocomplete) orthomodular posets. Characterizations of these orthoalgebras as well as orthocomplete orthomodular posets in terms of their chains are given. Also, a weak form of sigma -orthosummability, referred to as Weak Subsequential Interpolation Property, is defined for orthoalgebras; and it is shown that this latter class of orthoalgebras properly contains the former class of sigma-orthosummable orthoalgebras (called sigma-orthoalgebras). The Brooks-Jewett Theorem, the Nikodym Convergence Theorem and the Nikodym-Vitali-Hahn-Saks Theorem for semigroup -valued functions defined on a sigma -orthoalgebra (respectively, an orthoalgebra satisfying the Weak Subsequential Interpolation Property) are proved.