  Measurement in quantum mechanics is generally described as an irreversible
process that perturbs the wavefunction describing a quantum system. In this
work we establish a formal connection between the measurement description
within the Copenhagen interpretation (i.e., through the collapse of the
wavefunction) compared versus a picture in which the system and the measurement
apparatus are considered as a whole. We first consider a projective
measurement. In this limiting case, the natural requirements of consistency and
equivalence between the two pictures lead to the rigorous definition of
consistent measuring apparatus: the orthonormal wavefunctions from the Schmidt
decomposition of the system plus apparatus must have non-overlapping supports.
This result arises from the comparison of the two pictures (otherwise hidden),
and while it seems to be an obvious conclusion in the limit of projective
measurements, it has some nontrivial implications as one extends its validity
to the domain of weak measurements. In this respect, we argue on the existence
of two alternative approaches to mathematically constructing a weak measurement
protocol. While the two approaches are equivalent from the system's
perspective, they do strongly differ from the apparatus point of view, and
hence can be only distinguished one from each other in the picture where system
and apparatus are considered as a whole. We show that only one of the two
mathematical formulations of the weak measurement fulfills the consistent
apparatus condition, while the combination of the two gives rise to a
generalized weak measurements framework.
