Eigenstates into which a system can be projected after a measurement I'm currently reading Dirac's Principles of Quantum Mechanics, on page 36, he says: 

Another assumption we make connected to the physical interpretation of the theory is that, if a certain real dynamical variable $\xi$ is measured with the system in a particular state, the states into which the system may jump on account of the measurement are such that the original system is dependent on them.

On what physical basis can we make this assumption and why is it reasonable? 
 A: This phenomenon is called the collapse of the wave function. It is one of the tenets of the Copenhagen interpretation of quantum mechanics.
The eigenstates $|\xi_i\rangle$ of the $\Xi$ operator form a complete set. From linear algebra we have
$$I=\sum_i|\xi_i\rangle\langle \xi_i|$$
where $I$ is the identity operator. We apply this to the state vector $|\psi\rangle$:
$$|\psi\rangle=\sum_i|\xi_i\rangle\langle \xi_i|\psi\rangle$$
We have now expressed the state in terms of the $\Xi$ eigenstates. When we measure $\Xi$ and get $\xi_j$, we project the state vector onto the eigenstate using the projection operator $\mathbb{P}_j=|\xi_j\rangle\langle\xi_j|$. So after measurement we get
$$\psi\longrightarrow N\mathbb{P}_j|\psi\rangle=N\langle\xi_j|\psi\rangle|\xi_j\rangle$$
where $N$ is the new normalization constant.
A: When you measure an observable over a certain state, any possible outcome lies within the physical spectrum of the observable itself (which can be shown to coincide with the algebraic notion of spectrum for linear operators). So after the measurement has given you a value, say, $\lambda$, any other measurement on the system will give you $\lambda$ again with probability 100%. This is just the probabilistic interpretation of Quantum Mechanics, where the way the expectation value of an observable on a given state is defined as the average over a large enough ensemble of exact copies of the system in the given state. The outcomes must be in the spectrum of the observable, but they way the state is then defined gives you an average over all the possible outcomes. Clearly, once you have collected all the copies of the system in the ensemble that have given you the value $\lambda$, you now know that if you measure the same observable again on this subensemble, you will find $\lambda$ on all of them.
A typical example is that of polarisation. Once you have polarised light, the intensity through a polariser with the same axis will be 100%.
