In quantum mechanics, in general, it is stated that the act of measurement changes the state of the system. For example, consider the following Stern-Gerlach setup;
A beam of silver atoms first passes through SG apparatus in $z$ direction. The ones come out from the +z direction are led to another SG apparatus that is in the $x$ direction. The ones come out from the +x direction are then led to yet another SG apparatus in the $z$ direction.
Now, if we were to perform this experiment, we would see that the silver atoms having spin in the +z direction, initially, can have a spin in the $-z$ direction at last apparatus.Therefore, as one of my professor put, the act of measuring the $x$ component of the spin erased the information about the component of the spin in the $z$ direction.
However, in the above experiment, we made a selection at the end of each apparatus; namely, we only allowed atom having spin $+z/+x$ to pass through the next stage, so consider the following;
In other words, in this case, we allow the atoms coming out from $-x$ direction pass to the next stage.
In this case, the act of measuring the $x$ component of the spin will not "erase" the information about the $z$ component (this experiment is explained in the QM lectures in ocw.mit.edu), so isn't the statement "the act of measurement changes the state of the system" wrong in general ?
Moreover, if we think about it, almost all of the "problematic" (weird) things in QM comes from the fact that by making a measurement, we cannot determine the state of a system priori to the measurement because by making a measurement, we are disturbing the system, hence we cannot talk about the state "priori to the measurement" by doing a measurement, but the latter experiment says "well you can actually in some cases", so I see a contradiction in here.
Note that, in here, I'm consciously sending both output of $SG_x$ to the same $SG_z$ apparatus.