  It is a common view that von Neumann laid the foundations of a "quantum
probability theory" with his axiomatization of quantum mechanics (QM). As such,
it is regarded a generalization of the "classical probability theory" due to
Kolmogorov. Outside of quantum physics, however, Kolmogorov's axioms enjoy
universal applicability. This raises the question of whether quantum physics
indeed requires such a generalization of our conception of probability or if
von Neumann's axiomatization of QM was contingent on the absence of a general
theory of probability in the 1920s.
  In this work I argue in favor of the latter position. In particular, I show
that for non-relativistic $N$-body quantum systems subject to a
time-independent scalar potential, it is possible to construct a mathematically
rigorous theory based on Kolmogorov's axioms and physically natural random
variables, which reproduces central predictions of QM. The respective theories
are distinct, so that an empirical comparison may be possible. Moreover, the
approach can in principle be adapted to other classes of quantum-mechanical
models.
  Part II of this series will address the projection postulate and the question
of measurement in this approach.
