= Solution
In the <generalized measurement postulate>, outcome $m$ is described by a <Kraus operator> $M_m$, with $\sum_mM_m^\dagger M_m=I$. On a <density operator> $\rho$ its probability and conditional output, for nonzero probability, are
$$
\boxed{p_m=\operatorname{Tr}(M_m\rho M_m^\dagger),\qquad
\rho_m=M_m\rho M_m^\dagger/p_m}.
$$
For a pure input the conditional vector is $M_m|\psi\rangle/\sqrt{p_m}$. The effects $E_m=M_m^\dagger M_m$ are positive and sum to the identity, defining a <positive operator-valued measure>.
More generally an outcome can have several <Kraus operators> $M_{m\alpha}$. Then sum over $\alpha$ in both the numerator and the probability, and require $\sum_{m,\alpha}M_{m\alpha}^\dagger M_{m\alpha}=I$.
The usual <projective measurement> with the <Lüders rule> is recovered when the measurement operators are mutually orthogonal projections, $M_m=P_m=P_m^\dagger=P_m^2$, $P_mP_n=0$ for $m\ne n$, and $\sum_mP_m=I$. Projector-valued effects alone give projective outcome statistics; to obtain the Lüders state update as well, the instrument must implement $\rho\mapsto P_m\rho P_m$, without an additional conditional state change.
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