Observables in fuzzy probability theory. (English) Zbl 0984.60001

Gudder, Stanley

Demonstr. Math. 31, No. 3, 573-586 (1998).

This paper addresses an interesting development in probability theory, in which ideas on fuzzy, or unsharp observables originating from quantum measurement theory and operational quantum theory are carried over into classical probability theory: “Properties of observables in a recently formulated probability theory are investigated. After a review of definitions and terminology, we compare observables to probability kernels (or Markov kernels) and statistical maps which have been studied previously. We then discuss the spectrum of an observable, and a spectral mapping theorem is proved. Finally, a special type of observables called an apparatus is introduced and its properties are studied.” (Author’s abstract.) The paper is a continuation of [the author, ibid. 31, No. 1, 235-254 (1998; Zbl 0952.60002)], and it relates to work of S. Bugajski [Int. J. Theor. Phys. 35, No. 11, 2229-2244 (1996; Zbl 0872.60003)], and S. Bugajski, K.-E. Hellwig and W. Stulpe [Rep. Math. Phys. 41, No 1, 1-11 (1998)].

Reviewer: Paul Busch (Hull)

MSC:

60A05	Axioms; other general questions in probability
81P15	Quantum measurement theory, state operations, state preparations

Keywords:

fuzzy probability theory; effect; observable; probability kernel; spectrum

Citations:

Zbl 0872.60003; Zbl 0952.60002