When is the integration on quantum probability spaces additive?
type: Item
instance of: scholarly article
title: When is the integration on quantum probability spaces additive?
zbMATH Open document ID: 0734.28015
published in: Real Analysis Exchange
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 28C99
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 81P10
zbMATH DE Number: 4214298
zbMATH Keywords: additivity of integrals
zbMATH Keywords: generalized measure space
zbMATH Keywords: quantum logic
zbMATH Keywords: quantum probability spaces
author: Mirko Navara
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4214298