Collapse postulate in the density operator formalism

Question

To the extent that the collapse postulate holds, textbooks will almost invariably restrict the discussion to contexts in which states are represented by normalized kets, so that the collapse postulate reads (if $M_\alpha$ represents the measuring operation in which it is found that the result has (eigen)value $\alpha$) $$|\psi\rangle \to M_\alpha(|\psi\rangle) = \frac{\Pi_\alpha|\psi\rangle}{\left |\left|\Pi_\alpha|\psi\rangle \right|\right|}$$ How does this postulate generalize to the context in which states (even pure states) are represented by density operators (which obey the requirements of being self-adjoint, nonegative, and unit trace)? Does one simply first diagonalize the density operator and then "collapse" each part of the outer product in each term as above. That is, would one proceed as, where the expansion of the density operator is in its eigenbasis (is this even a necessary step), $$\rho = \sum_n p_n |\psi_n\rangle \langle \psi_n | \to M_\alpha(\rho) \stackrel{?}{=} \sum_n p_n M_\alpha(|\psi_n\rangle) M_\alpha(\langle \psi_n|) \stackrel{(1)}{=} \sum_n p_n \frac{\Pi_\alpha|\psi_n\rangle}{\left |\left|\Pi_\alpha|\psi_n\rangle \right|\right|}\frac{\langle \psi_n |\Pi_\alpha}{\left |\left|\Pi_\alpha|\psi_n\rangle \right|\right|} $$ where in (1) I've (handwaved?) and used that the projection operator is self-adjoint so that the bra related to the ket $M_\alpha(|\psi\rangle)$ is as I've given above. I suppose this also assumes that all projection operators in quantum mechanics are self-adjoint which I think must be true since observables are as such.