This is a common description of the measurement process and I don't believe it's correct. Where do we ever see a pure eigenstate come into existence as a result of a measurement? Let a photon pass through a pinhole so it spreads out. When it hits a photographic plate, it is absorbed. It is absorbed over the whole surface of the plate and ceases to exist. It does not resolve itself into a "position eigenstate of the photon". At the same time, a silver halide crystal undergoes an irreversible phase transition. That doesn't mean the photon got concentrated at the location of the crystal. The crystal didn't need the entire energy of the photon undergo a change of state: it was in a metastable state to begin with and only needed a small distrubance.
What about the photoelectric effect, where the escaping electron did indeed need the total energy of the photon? In that case the exact point where the photoelectron leaves the metal surface is indeterminate, and you can't say the photon was concentrated at a specific location.
What about an electron going through the Stern Gerlach apparatus? It divides into two beams: that is it's "state". When it hits the detector screen the two streams jointly excite a single bound wave function within the screen; at the same time, a crystal on the surface changes state. How do we know that the newly-excited bound wave function doesn't have the same spin that the electron had BEFORE entering the SG apparatus? There is no need to think that the electron, originally in a composition of two spin states, has resolved itself into one state or the other simply because a particular crystal in one branch of the wave stream has changed color.