= Solution
The <generalized measurement postulate> specifies <Kraus operators> $M_{i\alpha}$, where $i$ is the recorded outcome and $\alpha$ permits unobserved alternatives for that outcome, satisfying
$$
\boxed{\sum_{i,\alpha}M_{i\alpha}^\dagger M_{i\alpha}=I.}
$$
For a <density operator> $\rho$, outcome $i$ has probability and conditional state
$$
\boxed{p_i=\operatorname{Tr}\left(\rho\sum_\alpha M_{i\alpha}^\dagger M_{i\alpha}\right),\qquad
\rho_i=\frac{\sum_\alpha M_{i\alpha}\rho M_{i\alpha}^\dagger}{p_i}\quad(p_i>0).}
$$
With one operator per outcome, this is the usual formula $p_i=\operatorname{Tr}(M_i\rho M_i^\dagger)$, $\rho_i=M_i\rho M_i^\dagger/p_i$. The positive operators $E_i=\sum_\alpha M_{i\alpha}^\dagger M_{i\alpha}$ form a <POVM>, while the state-update maps form a <quantum instrument>.
The full postulate reduces to the usual <projective measurement> with the Lüders update when there is one operator per outcome and $M_i=P_i$, with $P_i=P_i^\dagger=P_i^2$, $P_iP_j=0$ for $i\ne j$, and $\sum_iP_i=I$. Then $p_i=\operatorname{Tr}(P_i\rho)$ and $\rho_i=P_i\rho P_i/p_i$. Projective effects $E_i=P_i$ alone ensure these probabilities but do not fix the state update: an additional outcome-dependent unitary could still be applied. The choice $M_i=P_i$ makes the required projective state-update condition explicit.
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