  The mathematics of classical probability theory was subsumed into classical
measure theory by Kolmogorov in 1933. Quantum theory as nonclassical
probability theory was incorporated into the beginnings of noncommutative
measure theory by von Neumann in the early thirties, as well. To precisely this
end, von Neumann initiated the study of what are now called von Neumann
algebras and, with Murray, made a first classification of such algebras into
three types. The nonrelativistic quantum theory of systems with finitely many
degrees of freedom deals exclusively with type I algebras. However, for the
description of further quantum systems, the other types of von Neumann algebras
are indispensable. The paper reviews quantum probability theory in terms of
general von Neumann algebras, stressing the similarity of the conceptual
structure of classical and noncommutative probability theories and emphasizing
the correspondence between the classical and quantum concepts, though also
indicating the nonclassical nature of quantum probabilistic predictions. In
addition, differences between the probability theories in the type I, II and
III settings are explained. A brief description is given of quantum systems for
which probability theory based on type I algebras is known to be insufficient.
These illustrate the physical significance of the previously mentioned
differences.
