  We present a reformulation of quantum mechanics in terms of probability
measures and functions on a general classical sample space and in particular in
terms of probability densities and functions on phase space. The basis of our
proceeding is the existence of so-called statistically complete observables and
the duality between the state spaces and the spaces of the observables, the
latter holding in the quantum as well as in the classical case. In the
phase-space context, we further discuss joint position-momentum observables,
Hilbert spaces of infinitely differentiable functions on phase space, and
dequantizations. Finally, the relation of quantum dynamics to the classical
Liouville dynamics is investigated.
