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Two-valued states on a concrete logic and the additivity problem. (English) Zbl 0597.28004
A $$\sigma$$-class C on a nonempty set X is a collection of subsets of X which contains the empty set and which is closed under complementation and disjoint countable unions. Probability measures and measurable functions on (X,C) are defined in the usual way. If f is measurable and m is a probability measure, then an integral $$\int f dm$$ can be defined in essentially the same way as the Lebesgue integral. The question considered in this paper is whether the integral is additive on two measurable functions whose sum is measurable. This question was first considered by the reviewer [Proc. Am. Math. Soc. 21, 296-302 (1969; Zbl 0183.287)]. It is known that, in general, the integral need not be additive. In this paper the integral is shown to be additive if m is two- valued, at least one of the functions has a nowhere dense range, and at least one of the functions is bounded.

MSC:
 28A25 Integration with respect to measures and other set functions 03G12 Quantum logic 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 06C15 Complemented lattices, orthocomplemented lattices and posets
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References:
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