Abstract
We generalize ordinary probability theory to those von Neumann algebras A, for which Dye's generalized version of the Radon-Nikodym theorem holds. This includes the classical case in which A is an Abelian von Neumann algebra generated by an observable or complete set of commuting observables. Via Gleason's theorem, this also includes the case of ordinary quantum mechanics, in which A=B(H) is the von Neumann algebra of all bounded operators on a separable Hilbert space H. Particular consideration is given to the concepts of conditioning, sufficient statistics, coarse-graining, and filtering.
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