When is the integration on quantum probability spaces additive? (Q810657)

In the paper in J. Comb. Theory, Ser. A 39, 42-51 (1985; Zbl 0624.28012) \textit{S. P. Gudder} and \textit{J. E. Zerbe} proved that for finitely valued functions the integral on quantum probability spaces (QPS) is additive. Their proof involved highly nontrivial combinatorial reasoning. By applying a new (essentially plane topological) method, we extend the additivity result to a broad class of functions. [Theorem 6: Let \((X,L,s)\) be a QPS and let f, g, \(f+g\) be bounded measurable functions on X. Let the ranges of f and g be nowhere dense sets in R and let the range of g be countable. Then \(\int (f+g)ds=\int f ds+\int g ds.]\) When specialized to finitely valued functions, we also provide a simpler proof of the original Gudder-Zerbe theorem.