I'm a little confused about the terminology of measurement.
So say that we have the single qubit state $|\phi \rangle=c_0|0\rangle+c_1|1\rangle$.
If we perform the projective measurement $P_0=|0\rangle\langle 0|$. Then we say that the probability of obtaining the measurement result $|0\rangle$ is $\langle \phi|P_0|\phi \rangle$. So in this case we're talking about a possible state that can occur in the collapse of the wavefunction of $|\phi\rangle$.
In the context of P.O.V.M. measurement where we assume it's not a projective measurement, and we label the measurement operators $E_i$ then we say that if the result of the measurement is $E_i$ then we assume that the state sent by Alice was state-x. The probability of the measurement result $\langle\phi_j| E_i |\phi_j\rangle$.
So, in this case, it seems as though we're talking about the measurement of an operator instead , but if $E$ can be projective operator and in the case of a projective operator we're measuring the probability of the state collapsing into some state, then why does it seem like that's not what's happening in the more general case ?