  A generalized Bloch sphere, in which the states of a quantum entity of
arbitrary dimension are geometrically represented, is investigated and further
extended, to also incorporate the measurements. This extended representation
constitutes a general solution to the measurement problem, inasmuch it allows
to derive the Born rule as an average over hidden-variables, describing not the
state of the quantum entity, but its interaction with the measuring system.
According to this modelization, a quantum measurement is to be understood, in
general, as a tripartite process, formed by an initial deterministic
decoherence-like process, a subsequent indeterministic collapse-like process,
and a final deterministic purification-like process. We also show that quantum
probabilities can be generally interpreted as the probabilities of a
first-order non-classical theory, describing situations of maximal lack of
knowledge regarding the process of actualization of potential interactions,
during a measurement.
