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Additivity of integrals on generalized measure spaces. (English) Zbl 0624.28012
A generalized measure space is a triple (\(\Omega\),\({\mathcal C},\mu)\) where \(\Omega\) is a set of \({\mathcal C}\), a \(\sigma\)-class of subsets of \(\Omega\) closed with respect to disjoint countable unions and complementation and \(\mu\) a measure on \({\mathcal C}\). While in the classical measure theory additivity of integral is an obvious and fundamental result, it is not so in the generalized measure spaces. In general the integral in this case is not additive. The question of the additivity in the case of simple functions is solved in the reviewed paper in a positive way.
Reviewer: T.Neubrunn

MSC:
28C99 Set functions and measures on spaces with additional structure
03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
81P20 Stochastic mechanics (including stochastic electrodynamics)
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[1] Gleason, A, Measures on closed subspaces of a Hilbert space, J. rational mech. anal., 6, 885-893, (1975) · Zbl 0078.28803
[2] Gudder, S.P, Elementary length topologies in physics, SIAM J. appl. math., 16, 1011-1019, (1969) · Zbl 0179.30203
[3] Gudder, S.P, Quantum probability spaces, (), 296-302 · Zbl 0183.28703
[4] Gudder, S.P, A generalized measure and probability theory for the physical sciences, (), 121-141, (Harper and Hooker, Eds.)
[5] Gudder, S.P; Marchand, J.P, A course-grained measure theory, Bull. acad. ser. sci. math. polon., 28, 557-564, (1980) · Zbl 0499.28002
[6] Gudder, S.P; Zerbe, J, Generalized monotone convergence and Radon-Nikodym theorems, J. math. phys., 11, 2553-2561, (1981) · Zbl 0467.60003
[7] Jauch, J.M, Foundations of quantum mechanics, (1968), Addison-Wesley Reading, Mass · Zbl 0166.23301
[8] Mackey, G, Mathematical foundations of quantum mechanics, (1963), Benjamin New York · Zbl 0114.44002
[9] Neubrunn, T, A note on quantum probability spaces, (), 672-675 · Zbl 0208.43402
[10] Suppes, P, The probabilistic argument for a nonclassical logic of quantum mechanics, Philos. sci., 33, 14-21, (1966)
[11] Varadarajan, V, ()
[12] Zerbe, J, Generalized measure theory, () · Zbl 0467.60003
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