Ideal quantum measurements
Fundamental measurement postulate
For an ideal measurement we have: Suppose some observable with operator $\hat{R}$ is measured on a quantum system $\rho$ Then the property $B$ is measured with probability (simplification with cyclic trace and projection property ) $$P(B) = \text{tr}{E(B)\rho E(B)} = \text{tr}{E(B)\rho E(B)}$$ and density matrix of the system after the measurement is $\rho^\prime$ $$\rho^\prime = \frac{E(B)\rho E(B)}{P(B)}.$$ Here $E(B)$ is the spectral decomposition of the operator $\hat{R}$ into projection operators with the property $B$.
Definition: Selective measurement
The original ensemble is split into sub-ensembles $\mathcal{E}_\alpha$ each corresponding to a certain measurement outcome.
Definition: Non-selective measurement
The different sub-ensembles corresponding to a measurement outcome mix. (I understand that for a given outcome state we can not certainly say the measured value).