Noncommutative Probability on von Neumann Algebras

We generalize ordinary probability theory to those von Neumann algebras $\mathbb{A}$ , for which Dye's generalized version of the Radon‐Nikodym theorem holds. This includes the classical case in which $\mathbb{A}$ 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 $\mathbb{A}=\mathbb{B}\mathrm{(H)}$ 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.