Then the results are that afterward:

For every value of $\lambda$, there is a probability of $|\langle\Psi|\Psi_\lambda\rangle|^2$ that the system is in state $|\Psi_\lambda\rangle$

This is captured by the density matrix formalism as the transition

$|\Psi\rangle\langle\Psi|\Rightarrow\sum_\lambda|\langle\Psi|\Psi_\lambda\rangle|^2|\Psi_\lambda\rangle\langle\Psi_\lambda|$

atyy I guess thinking about it classically, Demystifier's argument must be right.