Suppose $A$ and $B$ are compatible observables (i.e. $[A,B] = 0$). We take the eigenkets of $A$ to be $|a_1 \rangle \ldots |a_N \rangle$. Further, we suppose that the first $k$ eignekets of $A$ are degenerate while other eigenkets are non-degenerate. That is $A |a_i \rangle = a |a_i \rangle$ for $i = 1, \ldots, k$. But as is typically the case we suppose that the observable $B$ breaks the degeneracy and we may label the first $k$ eigenkets by their distinct $B$ eigenvalue. That is, we take $|a_i \rangle = | a, b_i\rangle$ for $i = 1,\ldots, k$ where $A | a, b_i\rangle = a | a, b_i\rangle$ and $B | a, b_i\rangle = b_i | a, b_i\rangle$.
Now let us say we have a state ket that is in a general superposition of the $A$ eigenkets: $$ |\psi \rangle = \sum_{i=1}^{k} c_i | a,b_i \rangle + \sum_{i=k+1}^N c_i|a_i \rangle . $$ We measure the observable $A$ and obtain the value $a$ (the degenerate eigenvalue). I have a few questions about this situation:
It seems that this should happen with probability $|c_1|^2 + \ldots + |c_k|^2$. Is this correct?
What state does the system jump into after this measurement? Sakurai (section 1.4) claims that the system jumps into a linear combination of the $|a,b_i \rangle$. What are the coefficients though? Are they the same as that in the original state but normalized to retain the overall normalization of the state ket? That is, do we have $$ |\psi \rangle \to \sum_{i=1}^{k} \frac{c_i}{|c_1|^2 + \ldots + |c_k|^2} | a,b_i \rangle ? $$
(Assuming that the assertion in 2. is correct) Would there be any experimental difference if instead we postulated that the system jumps into a particular $|a,b_i \rangle$ for some $i$ with probability $|c_i|^2$? For example, if all we did was measure $B$ afterwards we would get all the values of $b_i$ with the same probabilities under both postulates.