  A rigorous general definition of quantum probability is given, which is valid
for elementary events and for composite events, for operationally testable
measurements as well as for inconclusive measurements, and also for
non-commuting observables in addition to commutative observables. Our proposed
definition of quantum probability makes it possible to describe quantum
measurements and quantum decision making on the same common mathematical
footing. Conditions are formulated for the case when quantum decision theory
reduces to its classical counterpart and for the situation where the use of
quantum decision theory is necessary.
