  In this paper, we present a general theory of finite quantum measurements,
for which we assume that the state space of the measured system is a finite
dimensional Hilbert space and that the possible outcomes of a measurement is a
finite set of real numbers. We develop the theory in a deductive manner from
the basic postulates for quantum mechanics and a few plausible axioms for
general quantum measurements. We derive an axiomatic characterization of all
the physically realizable finite quantum measurements. Mathematical tools
necessary to describe measurement statistics, such as POVMs and quantum
instruments, are not assumed at the outset, but we introduce them as natural
consequences of our axioms. Our objective is to show that those mathematical
tools can be naturally derived from obvious theoretical requirements.
