  Theory of quantum measurements is often classified as decision theory. An
event in decision theory corresponds to the measurement of an observable. This
analogy looks clear for operationally testable simple events. However, the
situation is essentially more complicated in the case of composite events. The
most difficult point is the relation between decisions under uncertainty and
measurements under uncertainty. We suggest a unified language for describing
the processes of quantum decision making and quantum measurements. The notion
of quantum measurements under uncertainty is introduced. We show that the
correct mathematical foundation for the theory of measurements under
uncertainty, as well as for quantum decision theory dealing with uncertain
events, requires the use of positive operator-valued measure that is a
generalization of projection-valued measure. The latter is appropriate for
operationally testable events, while the former is necessary for characterizing
operationally uncertain events. In both decision making and quantum
measurements, one has to distinguish composite non-entangled events from
composite entangled events. Quantum probability can be essentially different
from classical probability only for entangled events. The necessary condition
for the appearance of an interference term in the quantum probability is the
occurrence of entangled prospects and the existence of an entangled strategic
state of a decision maker or of an entangled statistical state of a measuring
device.
