PeterDonis said:
It depends on what you mean by "measure the spin".
If you mean put a filter after the z-direction SG magnet, which only allows the "up" beam to pass, and then put that beam through an x-direction SG magnet and put a detector after it, then yes, the filter can be considered a filter measurement (or, equivalently, as the preparation of a z-spin up state).
If you mean just put two SG magnets in series (or three--your SG, SG1, SG2 scenario), with no filtering, then the first magnet is not a measurement, because there is no filter and no detector; if there were a detector after the first magnet, it would absorb the atom and make it unavailable for the other magnets.
If you just consider the magnet and not detecting the particles you have a preparation procedure for spin-component eigenstates (component in direction of the homogeneous part of the magnetic field), described by a unitary time evolution. The preparation is due to the entanglement (nearly 100% if you have a good design of the magnet and the incoming beam) between the spin component and the position (or momentum) of the particle.
I think in the usual sense that's indeed not yet a measurement, because with the preparation you have only established the correlation between the spin component and position of the particle, i.e., a correlation, telling you that if you register a particle in one partial beam you have with certainty "spin up" and if you register a particle in the other partial beam the particle has with certainty "spin down".
For an individual particle it is not predictable in which partial beam (and thus with which spin component) it leaves the magnet but it ends up with some probability in the one or the other beam depending on the initial state. In the original setup with a beam coming out of an oven you'll have probabilities of 1/2 for either outcome.
Of course, if you have just three magnets as described in the above arrangement then you just have to describe the interaction of the particle with the three magnets in the Hamiltonian to predict the outcome of the measurement after they went through all three magnets.
If you do a measurement after the 1st magnet or block one partial beam after the 1st magnet you have to describe this too, because then of course you change the state of the particle through the interaction with the measurement device or filter. Usually this is done in a FAPP description a la projection postulate, but in principle it's just an interaction of the particle with some material/fields making up the measurement device of filter and is described by the usual quantum dynamics (of the composite system consisting of the particle and the devices used to detect it), making the particle to an open quantum system. When "tracing out" the information about the state of the devices/environment of course the time evolution of the particle alone is no longer unitary.
A simple and in some sense extreme example is a Brownian particle in a "heat bath". Over some steps of "coarse graining" you end up with a (in general non-Markovian) Langevin equation describing the particle's motion or even just an equation of motion with some friction for its average position, i.e., a classical equation of motion. Then nothing is left from the unitary time evolution of the entire huge system consisting of the particle and the heat bath, but the derivation shows that this classical behavior is completely consistent with unitary time evolution on the full quantum level when considering the entire closed system, which of course you never can calculate in the full detail, because with around ##10^{24}## particles it's hard to solve the fully resolved dynamics of all these "microscopic degrees of freedom".